Not a punter/QB, but B. J. Daniels was a wide receiver and was also a backup QB for the Seahawks in 2015. So mixed roles still happen for backup QBs in the NFL. And why not, they’re taking a roster spot, get them on the field if they can contribute.
He currently is a QB for the Salt Lake Stallions in the AAF. (I know, who? But the AAF supposedly starts in February.)
Danny White was the Cowboy’s starting quarterback and punter in the same era. Tom Landry was the coach at the time.
He was, indeed, and I was thinking about him when I was posting about Pat McInally.
For his first few years in the NFL, White was the Cowboys’ backup quarterback (behind Roger Staubach), as well as being their punter. Staubach retired after the 1979 season, and White became the starting QB, while also keeping the punting job; he handled both duties for several seasons.