In fact just knowing you had the mulligan might make for better shots
I once played in a Pro-Am at the NLE Melrose-a course with multiple white knuckle tee and approach shots(especially in the wooden driver days)
In an unusual twist they allowed pros to buy one mulligan.
I was totally relaxed on every full swing, knowing I had a mulligan yet to use, and didn't miss a fairway and only a couple greens by a few feet.
On the 18th hole I hit a really solid putt from 8 feet for 67, that did a 180 at the hole-and I used my mulligan.
Now nervous as one can be, I missed the redo by 6 inches-I mean who can handle the pressure of only having one(more)try?