Director J.J. Abrams certainly does seem to have the golden touch when it comes to reinvigorating franchises, and almost any studio would be lucky enough to have him work on their IP.
Still, there is one property that the talented director doesn't necessarily see himself working on, which came up during an interview with CinemaBlend during Abram's press day for Star Trek Beyond. Abrams was asked if he would have any interest in working on the Indiana Jones franchise, and his answer might disappoint some people.
"I don't think so. First of all, they certainly don't need me. Secondly, I really do feel like, as someone who has been a very lucky recipient of other people's original ideas, and getting to sort of help shepherd them and be part of it, I'm looking forward to helping be a part of creating some stories that maybe one day someone can talk about rebooting. But it's been, with Mission: Impossible and Star Trek and Star Wars, and, frankly, now with Westworld too, which is a totally different thing, I do feel like it would be nice to work on something that doesn't necessarily pre-exist."
It makes sense that after helping so many franchises rise to the top, he would want to build his own empire, with ideas and creations that are exclusively his. What that will take the shape of is unknown and the director is still serving as producers on both Star Trek and Star Wars, so he currently has a great deal on his plate.
As for Indiana Jones, the next iteration of the series will see Harrison Ford return in the iconic role alongside Director Steven Spielberg, so it isn't like the film is lacking in standout talent.