Sorry for getting back so late. And, I know it's been discussed forever.....I searched it a long while back when I was a "newer" player.
For me, personally, there aren't a lot of shots I'd use this feature for. However, there are still a few. Road Hole bunker, for example. I'd like to hit a few different types of shots in there and what the effects of how the ball comes out instead of having to wait until next month when I finally hit another in there on accident. A "mulligan button" would allow me to intentionally put a shot in the bunker and hit a flop, push button, hit a pitch, push the button, hit a punch, and eventually see how the ball comes out each time to determine for the future which is my better default shot type.
This feature would be more for the newest players here. Those that don't know how to hit a flop or a pitch and are afraid to try in ranked play with friends or whatever. Those guys are out there. I used to be one.
Yes, it would affect scoring downward, and I know the common complaint here is that scores are unrealistically low. But, we are all striving for perfection and he who practices most will be the best most of the time.
I remember one of my favorite golf commercials of the early 2000's. Tiger Woods drove a golf cart up to a slope that looked like the inside corner of BPB's #6 dogleg, took out a bucket of practice balls, and just flung them from the bucket into the gunch. He grabbed a wedge and just went to work hitting them out individually.
I still wish I'd seen that commercial when I was 9 years old and playing everyday. I'd have immediately implemented that into my practice.