I think the more interesting question to ponder is not whether Bill wants to win another ring post Brady. Of course he does. But what would winning a ring with a rookie quarterback do for his career narrative?
When thinking about the other legacy coaches such as Walsh and Knoll, neither one of them won a championship without Montana and Bradshaw respectively. We all agree Bill is one of the greatest coaches, if not the greatest coach of all time but I think winning this year in particular with a rookie quarterback like Mac who is not a flashy or big armed quarterback, would immensely alter Bill's narrative. Much the same way Brady's narrative changed significantly after winning in Tampa Bay in year one away from Bill, if Bill could win this year, his first year with a rookie quarterback then I believe his legacy would be impacted just as much as Brady's was from a coaching standpoint obviously.
It would be a separator for Bill, 20 years after Brady, at age 70 to prove he could win again and do it with a rookie quarterback in a season where his team's expectations were not very high.