<insert obligatory "they should hire Greggles" comment here>
As mentioned above, this may be something to do with ex-Quizmaniac (I thought of that myself) Chuck Thomas being in charge?
I believe it might be to do with the News UK link, rather than Chuck himself.
This is all going off rough memories from the best part of a decade ago, so don't take this as concrete fact, but I believe Greg was pulled off air on his Sun Quiz Live debut. His first show was promoted for weeks, and on the evening itself, he had presented a few teasers live from the studio before he was due to go on air. About 15 minutes before the scheduled start time, the teasers stopped happening, and the show started about 30 minutes late with another presenter, and Greg's name was never mentioned again on any Sun Quiz Live broadcast.
I don't want to speculate on the reasons behind Greg never making it to air, but I'd assume that's why Greg can't do Q-Live.
Didn't he do a Sun Bingo thing a couple of years ago?