I don't think the real problem is about using a cover system
The problem, at least for me, is the poor implementation of cover based mechanics. They usually have these major flaws for me.
1) Lack of suppression. You can shoot all you want at your enemy, but he will still poke his head out and certain intervals. With the exception of the Brothers in Arms series, which when suppressed they just...poke their head out less and not as far. This ruins tactics when it really does just become like Yahtzee says, a shooting gallery. If they made enemies take cover...I dunno, from bullets that'd be great. You could then pin them, and move around them. Which I think would by itself make cover based shooting much more fun.
2) The cover is obviously placed. You will always have it. You'll always know combat is coming because you can only take cover on certain walls, that are about chest high. Levels end up looking more like cement bank queues than an actual place. I don't think cover should be designed into the level like that. What they should do it design a level that's believable, throw in some furniture or something, and make the player decide where to take cover. Or...maybe just not have cover in a room, so you have to hang out by the door and clear it out for a bit. It would change up tactics per room almost, rather than say...you need to run to that wall, shoot Steve the rent-a-cop, then move up to the next wall to trigger the next wave of security, shoot them all, move up to the next wall. You could make the flow so much better.
3) The lack of the ability to blind fire. Most games with cover tend to lack this feature. I don't think it's because they hate it, or they didn't have time. I believe it's not here because of the AI like in point 1. If you can't suppress enemies, there's really no need to blind fire. I think it could add a lot to a game having both of these.
4) Cover is absolutely necessary. If you come out of cover, the enemy has super accurate weapons, they don't breathe, and they have no heartbeat. They have bionic eyes, and super reflexes. You stop being in cover, ever bullet on this half of the planet is going right for you. I think when you sprint, you should be pretty hard to actually hit. You could use that to your adventage tactically too. (Sometimes it works in games...sometimes it doesn't. For me, more often it doesn't.) The ability to slide and such could be implemented too. (which seems to be coming in some games) Or bring diving back. Strangely getting to cover is pretty well done in the newer James Bond games (Quantum of Solace and Blood Stone)
...Giant wall of text, but that's really how I feel about it. I'd rather not have an alternative (because there aren't many really) I just think the current state of things need to be changed. The only games I think should kinda get noted for doing it properly (not perfect, but passably so) is Brothers in Arms, and Killswitch. Oddly both of them came before Gears of War...