I think the reason this game feels like it has more focus on cover than the old game is because there is no TU system for movement. And it's not necessarily that you used to end your turn out in the open in the old game but you often ended your movement out in the open, usually to have a look around or take a shot and then continue your movement getting into cover.
Now you basically have to end your movement in cover. You can shoot from behind cover and around corners, which is not a bad thing of course, but your choices are limited in the new game. You either move far into cover, move into cover then shoot, move to cover and then move to the next bit of cover, or shoot from cover and not move. Not being able to move after a shot in the last case seems like a poor design decision to me. However, there's the run and gun and hopefully more perks like that.
In the old XCom you could move, potentially find an alien, shoot at it and then move into cover. The new XCom makes you think a lot more about where you move your guys because you can easily mess up a move and not have a way out, which makes things interesting in a way. On the other hand your movement choices feel less dynamic.