The main complaint I have with cover based shooters is their sheer tedium.
I don't see why a shooter needs a mechanic to glue yourself to a piece of furniture/wall. You're given the ability to move in three dimensions and the ability to crouch/jump/sprint (in any real shooter), the combination of these is enough for all your cover needs. Having a cover system that promotes you peeking out and taking pot-shots slows down the gunplay incredibly and only serves to lower the skill ceiling, pad out gameplay, accommodate for controllers and mask terrible AI.
I don't find it fun at all to sit behind a piece of cover, completely protected (apart from a stray grenade), and then either blind fire or poke my head out to get some shots off. I find it dull, I find it slow, I find no challenge or worth to be found in it. Compare it to games such as S.T.A.L.K.E.R. or the classic shooters like Serious Sam where cover is manually taken and movement/repositioning is an incredibly important part of survival. Movement good, static bad.
Cover based shooting AND regenerating health is just the deathknoll of any shooter. It's the laziest way to design a shooter and also the most boring. It negates ALL threats and rewards you for doing nothing. Regenerating health already takes away rewarding exploration and repositioning, coupled with cover based shooting this is multiplied even more.
Now that's not to say that cover-based shooting can't be used at all. It's a useful mechanic for games which have very little combat or are more focused on story, having small segments of cover based shooting won't get too tedious if you use it sparsely. It can also be used if you're trying to mash up genres. Mass Effect worked as a cover-based shooter since the game was very RPG focused. Players had to control squadmates, items and abilities which the cover based shooting helped to give them some breathing room - something the later games in the series neglected, in my opinion.
I suppose my real main complaint is that cover based shooting is just used wrong and lazily which in turn makes the game dull and lifeless.
Ontopic:
Give me a god damn real-time/puesdo-real-time tactical cRPGs in the style of the Infinity Engine games. Stop crapping out shitty aRPGs and just do it. I don't want to "push a button and have something awesome happen", nobody but lobotomised rhesus monkeys want that.