BG is not required in order for UTA to be protected. I've covered that previously and believe that saving BG _only_ because of that _if_ we otherwise think Coffee scummy is a bad reason to not lynch him (of course, if you think he is town then that is different, but you're using this "we need the BG" angle).
We don't
NEED a BG to keep UTA protected, but as long as we
HAVE a bg around, we have +1 person protected during each night.
UTA only needs protection on every second night, so on odd nights, both the BG and the Doctor can heal different people.
For someone to be protected by BG, their role is disabled for that night, but UTA has no ability on even nights, which makes him a perfect Bg candidate.
Also, math and Coffee has equal chances to flip mafia*, so as long as they are both Schrödinger Cats, the only difference between them is role and activity, which Coffee wins both at the moment.
*the marked sentence does not consider scumreads, because both sides are hard to read, even though Coffee does have a bit more scummy vibes to him. These vibes are just not strong enough to make a relevant difference to me