Back in the '60s and '70s, it was not uncommon to see 3rd down FG attempts for just that reason. Teams didn't have long snappers, it was generally a job for one of the starting OL, and it wasn't always pretty, especially with muddy fields. Since the starting QB was often the holder, fake FG attempts on 3rd down were something to be watched for.
I'm not sure why teams don't use the backup QB as the holder. Is it because they think the backup QB's change so often there's no continuity? Maybe. I guess you'd have to look to see if teams change punters often these days. If so, there's no reason I can think of not to have your backup QB doing it - it gives you an option you otherwise probably didn't have.
I'd like to see Mikey Mike do this. make Hundley the holder, and at the exact moment in the playoffs, you pull off the fake and have Hundley roll out and throw the TD pass, or at least run for a first down.
Why not?
"The Devine era is actually worse than you remember if you go back and look at it."
KYPack
One good reason I can think of is the new rule that points after TD are kicked from the 15 yd line making successful fakes less likely.
I think another reason for having the punter be the holder is efficiency in practice. The punter and kicker can go off in a corner and do their thing without disrupting practice of the QB's and the offense.
One time Lombardi was disgusted with the team in practice and told them they were going to have to start with the basics. He held up a ball and said: "This is a football." McGee immediately called out, "Stop, coach, you're going too fast," and that gave everyone a laugh.
John Maxymuk, Packers By The Numbers