There is hardly a cutscene longer than a full song in the game (if you don't count Edensgate 1, of perhaps the Four Lord trials with their minigames). I'm sure you can find extremely niche examples but that's really most of the time not a thing, and when it's a thing, you can just reopen with everything because all the cooldowns are back anyway. You won't get a fully charged Apex perhaps, but even then it's not a given, because you can still cover 45s of an entire song over the cutscene.

Anyway, for more standard cutscenes, you just play around it. The way to do it is to make sure a song fully covers the downtime, which continues charging the apex gauge even if you can't do anything else. As long as you fulfill that condition, you'll not lose gauge generation and will be better off than many jobs that can't generate any resources at all. When it comes to songs drifting, maybe there is a case scenario that makes it awkward that I have yet to encounter, but most of the time you can just start the next song just before the cutscene starts and it should be fine. You can then resume with the next when appropriate after the cutscene.

Also i'm sorry to say it but I really hate the changes you're offering, they make songs completely irrelevant mechanically for the job kit on top of removing all flavor from them if they lose their own gimmick, which is already not that huge so I'd like to keep as much from it as possible...