No team with it head screwed on straight will permit Bill to take on GM responsibilities for their team. This does not mean some team might do so: there are certainly teams out there (us, for example, of late) who do not have their heads screwed on straight.
I do not think Bill would consider staying on as HC only, with a genuinely independent GM in place, but that is the only option I would offer him at this point. If he declines the offer, given growing concerns about his remaining expertise as HC, I will not be that upset. This team is starting over, from scratch, from its present status as a crap team. Why would one be upset over losing the services of the man most responsible for creating the situation?
I actually assume at this point that BIll will likely not be around in any capacity next year. My immediate concern is that he play no role in any plans or actions - trades, cuts, etc. - with implications for the team next year and in years following. I wish, frankly, there were a way of firing Bill the GM right now, or at least of undercutting his prerogatives in such decisions, starting right now.
This is not a "team with issues." It is a team which has been incompetently run into the ground. In important respects, it is the worst team in the league. It is, in a way not characteristic of most teams, a team run by one man. Any analysis of the way forward must begin there.