Never heard of memorializing a verbal agreement. Anything less than an NLI would have no binding effect on either party.
It would not surprise me if you see something like this in response to the new roster rules. Under the new rules you can only have 27 funded roster spots. But coaches obviously want, and need, more kids than that. Most will want 30-35 kids. They will "offer" non-funded roster spots to however many kids they want (max of 8). From the coaches standpoint, how do you make sure those kids actually show up? From a players standpoint, how do you make sure they don't change their mind at the last minute? Without an NLI neither side is bound in any way.
As you point out, verbal committments are pretty much a leap of faith for both parties. "Offers" of non-funded roster spots are going to be the same. I guess some sort of memorialization of the agreement might give both sides a little peace of mind, but that's about it. In the case you raiesed, maybe the coach thinks the player will be less likely to back out of his "committment" if he gets the kid to sign something.