The question shouldn't be about continuity or setting so much as if there should be another television series and who should be in it.
Having Star Trek on television dilutes it as a movie franchise to a certain degree. Why see a movie when you can see new episodes every week for free?
The first four movies generally made more money than the last six. TSFS, the least of the first four, made $76 million domestically. In 1984. Even without taking inflation into account, the only Star Trek movie of the next six to actually surpass that amount domestically was FC. That's it.
The less Star Trek is around, the bigger of a deal it'll be in the theater.
For argument's sake, though, let's say we did have another Star Trek series. Fine. It should not be anywhere within the vacinity of JJ Abrams' movies. They shouldn't be stepping on each other's toes. Whether that involves changing the characters, changing the setting, or changing the continuity, that's up to the Series VI to decide, but it would be better for the series and the movies to keep them separated so they can each fully grow into their own repsective entities.