I am a Star Trek fan through and through. I also really enjoy the JJ movies. Maybe it's the ability to look at things with an objective point of view that allows me to enjoy the JJ movies. I really don't like Enterprise. In my opinion it killed the TV franchise through multiple reasons, but I don't think that other fans who like Enterprise should stop claiming to be a Trek Fan. Everyone is entitled to enjoy whatever they want to enjoy.
I agree about some parts with what you're saying about Star Trek: Enterprise. Their heart was in the right place, they just unfortunately killed the show. Season 4 of Enterprise was starting to get a little bit good and the ending for the show with Commander William Riker & Commander Deanna Troi on the USS Enterprise NCC-1701-D just killed the show for me, it all being a holodeck simulation, worse ending ever to a Star Trek show.