I agree with you. We would have surely been told at the time that whoever won UBB would also become the new presenter if it ever came back.
No, not really, because Channel 4 tried to bury it in the ground and say it was the last ever series, trying to persuade us that it was never coming back, and then a year later it returned on Channel 5 with Brian as host. Channel 4 didn't want anyone to take over it, they just wanted to end it so they did their whole 'BB funeral' thing.
Besides, I doubt Davina would've said on the very last episode "Oh, Brian, if Big Brother ever comes back on a different channel then you'll host it!" because A) They wouldn't want to advertise for a rival channel and B) They were attempting to finish it off completely and there was no mention whatsoever on BB11 or UBB that there was a possibility of it returning.