I never had a problem with Enterprise being just "Enterprise," but I guess I'm the only one.
I never did either, but that's because I'm a hardcore fan and to me it doesn't matter if Star Trek is in the title or not, I'll still know about it. I think the people promoting the show thought that maybe regular people weren't watching the show because it didn't have the brand name attached to it, so they added it back on. In actuality, I don't think that was the problem.
I mean, do people not know that Gotham is about the world of Batman? Does that show's ratings suffer because it's not called Batman: Gotham (regardless of if that even makes any sense given the lack of Batman)? I'm guessing it doesn't make a difference, especially in this day and age.
When Enterprise hit the stage, people weren't as tuned in to entertainment media and the internet as they are now. If they missed it, it's because it didn't show up in their newspaper. People aren't going to miss it if it were called Starfleet or some other such thing. People didn't miss Better Call Saul because it wasn't called Breaking Badder: Better Call Saul.
Picard and Data are just as much "Star Trek" to me as Kirk and Spock, if not more given who I grew up with.
That's fine. I'm not saying Picard, Sisko, Janeway, or Archer aren't Star Trek, just that they aren't the embodiment of it. They're spin-offs. The 2009 movie wasn't.
At the moment, I can't think of any spin-offs that just used the same name as the original. They always add or change something, even in today's age. It's not very likely to just be "Star Trek."