Baseball does an adequate job with its system of minor league teams. Pretty much every other country in the world manages to have professional sports leagues without having universities be their feeder teams.
Even in your scenario, there's nothing stopping the aspiring golfer from attending Stanford part-time while they try to make it in pro-golf. Or applying and deferring admission for a couple of years. Frankly, they'd do better academically if they didn't have golf distracting them.
the minor league system pays significantly less than poverty wages. incentivizing kids to go that route instead of getting an education is terrible.
which is why so many dont, and go to college on scholarships anyway, which leads us right back where we started.
the right analogy to make IMO is ice hockey, especially the canadian model. There, you have well funded and professional-like "minor leagues" (called juniors close to same thing). you can't get paid but in the tier 1 leagues you have very little if not zero personal living expenses, it's like being on a full-ride at a big college program with room and board included. In canada, it's seen as the route potential NHLers take instead of continuing with school. you're not allowed to be paid, but there's a rule in place that for every year you play in that league you're owed one full year of college paid for. If you age out of the league and dont make it as a pro, you at least get 2-4 years of free college to go get the rest of your life together as compensation.
The US Junior system has some of those attributes but is more geared towards getting kids into NCAA scholarships. The thing there though, IMO, is that the money involve in that sport is right about at that sweet spot where i'd argue a full ride to school like notre dame or penn state is about right for the level of compensation owed.
all opinions above are my own, in the context of a former D1 ice hockey athlete who thinks the football players should probably be paid.
in minor league baseball, who's indepent teams may be anywhere on the spectrum of losing or making money but who are also affiliated with parent organizations that make hundreds of millions, there is a plethora of literature lately that accuses them of paying significantly less than US minimum wage. In many cases it appears the players pay them for the right to play when it's all done.
in the college hockey example i referred to, the D1 teams/leagues usually have decent revenue from tickets and tv deals but it's not a ton and i'd take a shot in the dark that most programs are pretty much breaking even after scholarships, staff, and facilities are accounted for. In that environment, having an athlete get "paid" maybe a $40k/year total package in scholarship, room and board, playing equipment and facilities use, and cash per diems on road trips is both above minimum wage and probably no more than they could reasonably expect if the league was "pro" and had nothing to do with college.