Finished my playthrough on Expert yesterday. I really enjoyed it. I don't think I would like to see the mainline XCOM series adopt any of the new ideas here, but as a spin off they work really well. I think the rest of this post is going to end up with a critical tone, but please take it as quibbling about something I really liked rather than anything else.
Biggest criticism: the difficult is off. Particularly single encounter missions later on are a complete cakewalk. The balance on once-per-mission abilities makes them too powerful if an encounter is all of the mission. Even the final missions were too easy, a maxed out crew is too powerful. I regret not putting it up to Impossible for the final third.
The timeline I don't mind too much, but it does push you into a very artificial feeling strategy of choosing your targets based on when they act next. I felt there wasn't enough stuff that interacted with the timeline (or perhaps I just didn't have the characters that got those abilities?), but those abilities that did interact it felt super powerful so perhaps that was necessary.
The biggest problem was the combination of the small rooms and the ability to spam character powers meaning that most of the time I kept my entire squad more or less in place unless the mission forced me to do otherwise: taking a second shot, lobbing a grenade, or healing a party member was almost always better than moving. This led to very static encounters. Whereas in the XCOM mainline you want to flank and outmanoeuvre foes, here I was mostly just staying in place and blasting away. The fact half my team had ignore cover abilities made this even more tempting.
All of this gave the game a refreshing new feel - which I think is what they were going for - but, ultimately, felt to me like a structure that offered less than the traditional system. Mind you, perhaps if there is a Chimera Squad 2 they could iterate on all these systems and improve the way they work as well.