TGC's original charter was to have cross platform play and cross platform course sharing. They're saying the reason they can't do cross-platform play at this point is because of the gamertag issue. One platform's name restrictions are different than others, so a PS4 gamertag might not be legal on XBox Live, and vice-versa. HBS says the way the username restrictions work right now, the best they could do would be to have you playing against someone called "TGC Member" in live or ghost ball play. That would really narrow it down, wouldn't it?
I think they'll eventually figure out a way to register TGC players names such that their platform's gamertag is somehow connected to a centrally registered TGC member name. That TGC membername could be formatted in such a way that the player had their personal tag with a designator at the beginning or ending of the TGC member name that would indicate what platform they were playing on. There will be some who will be playing on more than one platform and you'd probably want to know what platform they played a given round under.