The likeliest scenario for a "fall of the Federation, even though they've got shiny time-travel tech" is that whatever it would have taken to prevent the fall would have resulted in different people running the show, something that those in power weren't prepared to listen to as they'd rather have kept their power, even in a declining Federation than not had it in a thriving one.
How can we be sure it wasn’t because of that shiny time travel tech the Federation collapsed?
