I feel like responding to these two posts for fun, not arguing pro or anti asari overall though.
Low Birth Rate - Indeed. I get the sense that multiple children happen, but they're over the course of centuries and they're more planned than any other race (except perhaps Salarians). The pre-Council age of Asari may have had more numerous children but it seems that the gist for a while now has been a mix of having fewer and planned children, and mingling with other races, and to an extent welcoming other races into their territory more than others would (knowing that any children produced will be asari). They could steadily dominate in population if they tried, but they do not consider that an intelligent course of action. There are the theories that asari leaders do wish to take over the galaxy bit-by-bit in births and absorbing societies, but that remains anti-asari theory.
Weak Tech Innovations - Actually, the gist I got was that asari are the overall betters, but their innovations are also just, well, overall ones. That they're on the vanguard of generally used tech, but otherwise get beat by others. The turians go for military applications. The salarians go for higher level of tech (they experiment more) seeking an edge; this does not mean being an overall better. The quarians have ships and synthetics. And I get the sense that the humans are like a lower level of asari - there's lines that almost imply that, like Miranda speaking of asari society as something to aspire to. Though while asari look long game, humanity may upstage them in looking at the short and mid game. It may be the ME3 war that ends up putting humanity over asari but that's very up to opinion, and asari may still very well continue to get their overall edge. Don't look for them to as often go "I have this solution for this thing right now!", but they may, especially during peacetime, have a consistent stream of improvements down a pipeline.
Don't Heed Reaper Warnings/Thessia Didn't Hold Out Long - I think there's the official face and there's the unofficial face. The official face denies, but would have to admit that something is up. The unofficial face plans and figures things out, but not to the extent of actually believing in the Reapers. I think the Council races, all of them, knew they had to keep society, economy, military running and ready for anything, but it wasn't most often to the extent of being ready for the actual Reapers. So sure, the asari didn't heed Reaper warnings, but they didn't completely stick their heads in the ground either - just possibly more than the other races. I think they seek stability over all, and the extent of the Reaper threat is so unstable that they were the most in denial about it (though the Turian Councilor denied it more, it may have been more in his personal anti-human sentiment). So agreed, they should have been better prepared. The big main weakness of the asari is, apparently, their belief in overall superiority (even moral, planning, etc - not just for combat). A superiority that seems to be, most often, in most ways true, but in itself is something that can be exploited and having them panicking.
And agreed - it may have not been shown on screen, but the asari had been fighting the Reapers since the first half of the game. Reaper forces headed for at least the fringes of asari space since hitting Palaven, keeping them busy, so when the asari councilor says that they need to secure their borders, while we may be Angry Shepard about it, its technically a very solid plan for a TYPICAL sort of invasion, and we're just mad because we have a history of them minimizing our concerns, dismissing much evidence we have, and rejecting our experiences. Anyway, during the Rannoch arc the asari are well into the fight and Thessia itself is being threatened, and then its attacked and we have a period (Sanctuary, Other Stuff?, Cronos) where they're in the fight as much as anyone else.
Asari were ready to protect their territorial borders but they lost it so they had relatively less defenses after that (no one had ever breached in such a way before - again, a hubris of their supposed sense of superiority). They were also the best to take on husk forces overall, but the Reapers figured that out and probably swarmed them the most of anyone they invaded. In a way, they kicked ass better than at least most, but the Reapers apparently had enough of them and crushed them hard. Its like how asari commandos are the 'best'. Sure they're the best but there's also not as many of them and as commandos there are various ways to exploit their organization and methods.
Hid Knowledge of Prothian Artifact - I partially disagree. In the politics of the asari, the Matriarchs of Thessia are indeed in a whole (albeit fluid) class of leadership and I tend to think that counts enough. It may have been a faction within them that knew of this, but it was enough for the councilor herself to say "highest level of our government". Its not the whole people, but it IS their political leaders. Is Hackett the Alliance? I'd friggin say so. No matter their reasoning, the story of ME3 has it as the asari republics being liars and criminals about the laws on prothian technology. Especially since asari government is rather fluid and ought to have caught on to, exposed, and fixed the situation of hoarding prothian information. A flaw I could claim of the asari (though also everyone else) is hypocrisy. This all said, this isn't a matter to rationally demonize the asari over. There's a lot of context about it that makes clear that, as you said, the knowledge was very exclusive. We can call the asari to be more sneaky than they present to people (especially with info over the trilogy about stuff like AY and the refuges for them), but not evil or worse than anyone else. Just not perfect - unlike most of their portrayal in ME1 save for some cranky NPCs and Benezia being indoctrinated.
Lifespan - Agreed. They're the outright best here. Could be surpassed with medical tech, but in the trilogy they live the longest.
Biotics - Again agreed. Another example of 'overall best'. There may be misc others that are better than the average asari, but they're misc. Humanity/Cerberus seemed to be trying to find ways to get biotics that is at least on par with asari though, and not wait decades/centuries+ to maybe get it to happen. Many asari don't practice their biotics just because its a normal skill to them that isn't a big deal. We may see the world through a soldier/agent/war lens more often than not (even in the locations we visit), but most asari don't, and many see biotic practice as essential as we see swimming - aka helpful and probably easy enough, but not necessarily essential. But almost no asari would have a barrier to biotics, yes. Any amps for them would be the bonus, not the required tool.
Melding - Yeah the whole melding stuff is probably very useful. It communicates in a way that isn't very concrete, so not often as good as sharing computer data, but it can convey messages and feelings and genetics and probably even affect the way each other sees the world. 'Knowledge' in a general sense. Other races do not have this. Humans might be starting to gain this if theories about Shepard are true, but again like biotics is just not at their level overall. Drell retain 'perfect' memories but that doesn't have to do with transmitting anything. Hanar communicate on a atypical wavelength but it isn't as deep and personal as asari can when they meld. The asari may not be a more literal hivemind, but they do share qualities of it that maintain a strong integrity of their species, and having more asari be more like other asari, something in cases that would be a rather useful quality, than humans would be like other humans. They're rather more unified than others are (even if the trilogy takes care to show exceptions), and their melding ability is probably a big part of that.
Maiden - To be clear, exploring bars on other planets, making money off their sexuality, and getting a thrill from joining mercs is still something that qualifies the description for the Maiden stage, just as human male teen/youth competitive streaks can be expressed through physical violence or rising the ranks of a company. We can be sure that most maidens do not take this more seedy path, but ENOUGH do that it is a CONCERN to those paying attention, or especially watching it all from Illium as a bartender or with experience dealing with Omega
. Think of it as a very extended rebellious teenage phase that even includes running away from home, while most other teens conform enough to stay at home and do their studies. Sure it would have been nice to see more of the peaceful asari lands but the portrayal so far isn't something I'd consider broken. They do make the points in ME1 of having Liara be more innocent, ME2 where Illum is still a sign of how relatively peaceful asari space is, and ME3 where we get all sorts of stories about Thessia etc.
Fast Cultural Adaptation - Yeah, they're relatively quick to adapt and yet do it slow enough to not bring the accusations of appropriation and domination. This is one of their biggest pluses on the galactic stage, and a source of the theories that the supposed peace of the asari is in fact just a very very non-violent invasion 
Few Wars - Totally agree. Though they do seem to have the downside of being unable to win a war alone. We don't know of pre-Council asari much, but the others had wars internally and finished them, and the Council era wars also got finished by a side that was arguably much more of non-asari than asari. It can be said that the asari are just so good at ducking and weaving around actually both starting and ending wars, but it can also be said that it was this quality of them that had them seem to be relatively useless against the Reapers, as a society overall. It certainly didn't seem to be the asari that gathered the races against the Reapers this time, at least (unlike the rachni and krogan).
Biggest Economy - Definitely. It may compete heavily with a bunch of other economies and increasingly the humans, and the volus may have disproportionate economic influence, but the asari seem to have the biggest presence overall. Anyone else, even the volus, bowing out could still mean there's an economy, but I feel that the asari disappearing would basically mean everything is shattered and would have to start from scratch.
Best Military - Agreed but also disagreed. Disagreed because I do think that if the ONLY asari matched off against ONLY the turians, the turians stand a damn good or maybe even better chance of winning. But agreed because there's almost no chance that the asari would go solo into a war, whereas the turians look like they'd be content to. There would almost certainly be an alliance of asari and someone else. Its easy to imagine all sorts of war simulations done by the turians to see if they could in fact defeat the asari, and finding that the asari would almost always have the salarians (and/or more) and turn the tide, so even the most expansionist and independent turians just go "Okay... we'll be content with being on the council and holding more influence than the others on some matters.", ha.
I think if the asari focused on military growth as much as the turians do, they'd beat anyone, but I think they ultimately think longer term than that and are okay with being considered 2nd on that matter, as long as they have the confidence they'd win the actual conflict through their own means.
Then of course the krogan arguably had the best military in the past but they also burned out, even if they would argue it was through very devious and cheap means (genophage).
So basically I mostly agree except on a few points.
Are the asari the master race? No, and the story is a bit about that. But they were at least a supposed existing apex race until the Reapers invaded or Humanity rose, and either of those things were ME3, not ME1-ME2. That fits into the theme that became pretty big by the end of ME3, of apex never really being apex, and of chaotic surprises always being possible.
I describe the asari to others as 'better than most of the others in most ways, but exploited hard by some of the others in some ways'.