First thing is the immunity argument. You're perfectly right to say that no faction is immune to everything. Where I tend to disagree with you is that you don't lose any DPS over it because there's always another skill that can cover.
OK, you got me there. His DPS drops off a bit, that is 100% true, but it is nowhere near what most people imagine it to be.
The reason I think it is mostly psychological is this part right here.
The Qunari example is very nice, but you deal most of your damage with Lady's Wrath. Crushing Leap and Hakkon's Charge are gimped by immunities left and right. If all 3 abilities were doing decent damage, you'd be doing a lot more sustained DPS. Alas, that's not the case.
Yes, some enemies are immune to Crushing Leap. Luckily, they are mostly the larger "elite" enemies. The weaker cannon-fodder type enemies are not immune to fire, and they are the type of enemies that swarm en masse, which is exactly what Crushing Leap is suited for. The cooldown time for it is about the same time it takes for loads of enemies to spawn in and swarm you.
Yes, some enemies are immune to Hakkon's Charge. You can still knock down Spearmen, then hit them with either Crushing Leap, Lady's Wrath, Auto-Attack, and they die very quickly from all those things. Shock-Troopers dies incredibly fast from Lady's Wrath, so whatever.
Yes, Lady's Wrath deals damage to all enemies, but that doesn't mean it deals most of my damage. All three abilities deal good damage. I feel like people just write off the ability that the single enemy they are focusing on is immune to, and forget that all the enemies around them are still weak to the other abilities.
- Just because you are fighting some Spearmen/Shock-Troopers doesn't mean you should ignore Hakkon's Charge when it is available to use. There are likely Agents/Defenders/Assassins/Saarebas' around that you can hit with Hakkon's Charge and deal its full potential damage to.
- Just because you are fighting some Defenders/Shock-Troopers does not mean your should ignore Crushing Leap when it is available to use. There are likely Agents/Spearmen/Assassins/Saarebas' around that you can hit with Crushing Leap and deal its full potential damage to.
Not only that, but you have more than these three abilities. If an enemy is immune to fire damage, you are not actually stuck to two other abilities with longer cooldowns. You still have Korth's Might, which can still deal cold damage, electrical damage, and physical damage, as well as auto-attack. You assume you have lost a third of your offensive capabilities, or that you only have two abilities to do damage with, but in reality you still have six of your eight avenues of offense open to you (if you include auto-attack).
Now change faction to Red Templars and Lady's Wrath is basically useless with all the lightning immune Horrors running around.
True, Lady's Wrath is useless against Horrors, but Hakkon's Charge can one-shot them on Nightmare. If you didn't hit all the Horrors with a nicely lined up Hakkon's Charge, or if you miss part of your charge, you can follow it up with Korth's Might, Crushing Leap (all the Horrors are likely dead at this point, but just in case they aren't), another Korth's Might, and yet another Korth's Might. All of those can be done back to back with no wait time for cooldown, which can probably then be followed by more Hakkon's Charge, if by some horrible miracle there are still Horrors around. The whole time you do this, other enemies around you are being hit my Lady's Wrath, Crushing Leap, and hopefully by Korth's Might and repeat Hakkon's Charges if you have good positioning going on. Again, most of this should be unnecessary unless you have low crit chance/no Hakkon's Valor.
Change the faction to Venatori and the mages are annoying as hell by being immune to a different element each round.
Yep. This is the Achilles Heel of the Avvar. Spellbinders actually do make it as hard to do damage as people make it out to be. You cannot use the ability that the barrier is immune to, because nothing around is weak to it. For this reason alone, I hate Venatori. On Heartbreaker it's even worse with the Mages constant CC and Fade Cloak. 
The last thing I'd like to point out is that solos are a bit less representative. Not that it's not a very valid situation, but in a 4 player team the enemies aren't swarming around you all the time, they move around your teammates too. Spreading the enemies has a tremendous impact on how much "total" damage you can do with AOE abilities.
It's actually a perfect representation of how I play in a group. I rush in before anyone else can aggro the spawn so that I can nuke them all right away. Pugs probably hate me for that, or they love me if they are the type that shouldn't be in Nightmare in the first place. I don't know. Whatever.
If your team is pulling to a corner, his AoE capabilities are still very useful, not only because they pack up on the corner/doorway, but also because of your abilities to hit through walls.
Maybe it's just my perception, the Avvar is still perfectly fine in my book. Personally, I'm much more aggressive with my Katari because I can spam my abilities in a 6 second rotation. I can't do that with my Avvar and I do see a difference. I'd like talk a bit more about it, but I need to go as real life is calling for now...
I haven't played the Katari much since I have acquired my Valor, so I can't really say if I can I would play him more or less aggressively than the Avvar. The Avvar can't spam all his abilities in a six second period, but he can spam a rotation on Hakkon's Charge, Crushing Leap, Lady's Wrath (with an uptime of 80% of the time), Korth's Might in between all the other abilities, plus the extra non-elemental smash. By the end of rotation, usually you can start all over again, with the exception of Crushing Leap with a higher cooldown, so the rotation isn't perfectly smooth.
TL;DR - All abilites still do full damage due to enemies on the periphery of your main target still being vulnerable to abilities the main target is immune to. Use situational awareness and knowledge of enemies elemental immunities/vulnerabilities to know when to use abilities and which enemies to attack with said abilities. Do not discount Korth's Might, seeing as it can deal damage from all elements, as well as apply elemental status', as well as physical damage, in an AoE.
Also, I am trying not to come across as trying to "convert" you to the Avvar. I am just trying to give some tips, clear up misconceptions, etc. If you feel like I am coming across that way, or that I am in any way sounding like "L2P Scrub/Noob" just kindly tell me to shut the **** up and I will take no offense. 