I have friends (and met electro bands) who use Ableton Live for exactly what you are wanting to do.
Never used it myself but I am impressed by how flexible it is
A very clever jazz bass, sax and guitar playing mate uses it to record loops on the fly and then improvises over the loops.
Logic is very powerful for recording and mixing - a studio workhorse, but from what I have seen Ableton seems to be much better suited to live performance work where the structure may change or you may want to quickly extend the groove for a further 16 bars or whatever if the audience is digging the tune.
It has always been able to to all the real-time tempo and pitch matching magic with loops as well which blew my mind when I first saw it in use.
I'm a dedicated Logic user, but I'd have to say that Ableton would be the way to go for using as a live performance tool....Mainstage was developed to meet the challenge, but from what I have read it still doesn't quite cut the mustard compared with Ableton Live.
[just my 0.02]
I don't suffer from insanity; I enjoy every minute of it.