If I could enter this from the PoV of someone who didn't get ME1 or ME2 until long after all the DLC had been released (last August to be precise).
I always do LotSB before the SM, but after reading this thread have left the Normandy tour until after the SM to get the more flavourful conversation. Upgrades to 4 weapon types and resources/locations of resource-rich planets as well as the investment options, advanced training and research facilties are pretty huge in my opinion, so to say Liara doesn't help Shepard with the SB resources if you do LotSB pre-SM is 100% wrong imo.
As for drawing on experience of the real world release schedule, I can see how that would colour your interpretation, but I got ME1/ME2 as a package, played through vanilla once and was so engrossed I ended up paying for/downloading ALL the DLC (including, sigh, Pinnacle Station) in one afternoon. Even got a friend in the States to buy/drink (poor sod) some Dr Pepper so I could get the 3 hats. Once I'd played all the DLC with a post-SM character over the course of the next day or so (stupid work) I felt like something was wrong, like the Suicide Mission really should be the culmination, so I played the DLC once my 2nd Shepard hit 30 and the loyalty missions were done, and that felt right.
The way I see it is this. Liara openly warns Shepard that her office is under observation, that Illium is not a safe place to have an intensely sensitive conversation (and we're mainly talking SB/Collector stuff here) in response to Shepard's "tell me what's really going on". At this point Nyxeris is still around and standing right behind you as well potentially 2 Cerberus operatives (depending on who is in the party right then).
Shepard isn't stupid, being N7, Spectre, all that stuff that you only get with serious amount of grey matter. Hidden geniuses aside, how many of us are going to be that smart? Shepard gets the espionage angle IMMEDIATELY and accepts that there are some places the conversation cannot go, but yes, the heart will be pulling strongly in a direction the head knows it shouldn't. I can't reconcile the kind of lovestruck teenager behaviour that some folks seem to advocate with the character of the (wo)man we're talking about here. The mission comes first, but a few minutes wandering around Illium hacking the odd terminal isn't going to hurt. Taking days/weeks/months out to chase down a lead IS. Which is why I only do LotSB after all the other loyalty missions and after Overlord/Arrival.
Liara's had 2 years to mourn, Shepard still loves Liara with the intensity of only 4-6 weeks' elapsed time and has NOT had to mourn Liara. (S)he knows Liara still cares, the kiss shows that much, but also that Shepard's visit terrified her. How many times have any one of you really, truly believed someone who says "it's all right, I don't hate you"? To the extent that all that fear just evaporates immediately? I certainly haven't. Shepard would feel constrained to give Liara some space and time, and just be there for her. I think even a Paragon would help Liara out on the odd shady deal, just to show her that yes, the love is still there.
To quote Captain Carrot - "personal isn't the same as important". Shepard deals with other peoples stuff first, then goes to Liara with this supposedly helpful piece of info. I tend to do loyalty missions in order of objective importance, so Jack's tends to be last (dealing with leftover issues from years ago), things like Samara's, Thane's, Grunt's (all life-threatening to either the crew member or innocents caught in the crossfire) will take priority.
Shepard handles that stuff first, deals with Hackett's mission and gets a stay on the trial/surrender because the Alliance is losing ships to the Collectors and they know Shepard's their best hope, visits that strange AI project and gets his crew as ready as they possibly can be. Following that line of reasoning I end up with about 4 or 5 side-missions on random planets to do after LotSB, purely so I can trigger the upgrade terminal and save/load scum until I have all 5 guns at 6/6, the only time I really allow metagaming to colour the way I play through.
Once the crew are happy and daddy issues are sorted, Shepard indulges him/herself with a quick information delivery run to Liara in the hope that by giving her some space she'll be ready to talk about the relationship a bit more. Of course, as we know, it turns out much much more involved than that. My interpretation of the thought process Shepard would go through in LotSB.
1) The SB's only previous involvement with Shepard was providing the info that lead to Tali in ME1 and accidentally providing a squad member in Wrex. No reason to hate there.
2) The SB wanted to sell Shepards body to the Collectors. Well Shepard's now alive, has a good strong ship, crew and squad. Not much chance of that now, no need to make it personal.
3) The SB is trying to kill Liara and is holding the person who helped her save Shepard's body to allow him/her to be brought back. The SB is known to still be in league with the Collectors and, big revelation, is shown to have massive numbers of mercenaries (rather than just hiring Wrex), doesn't give a crap about civilian casualties any more (blowing up Drakon Trade Centre) and has at least one corrupted Spectre on the payroll. These facts are all revealed before you leave Illium and commit to taking the Normandy anywhere.
tldr: if you didn't get each bit of the DLC as it was released, I feel it makes sense to go loyalty missions > overlord > arrival > LotSB, finish up upgrades, gogo IFF/SM.