I guess you can make the case that a fitter golfer is a better golfer, but, you don't have to be fit to be a good golfer. I, personally, know a fat, out of shape smoker who is a scratch golfer.
There's simply no point in the game where someone would find himself in oxygen debt.
I agree, but 9/10 those are not one's walking the course. Gary Player was the first golfer to bring fitness into the golf game. Many people thought (and some still do) that just because you played a game that was deemed non-athletic that you didn't need to be really that fit and of course lifting weights was out of the question, much like when baseball coaches were telling players to not lift weights for fear of them getting too big and losing flexibility.
Now a days, if person was really serious about their golf game they will hire a trainer not only to add more power and flexibility but to reduce the risk of injury. If they can add more power it translates into more club head speed along with the improved flexibility. 90% of these young guys on tour are, in fact athletes and they freaking train like athletes and the health benefits are off the hook. If you follow golf like I do you know the story of Eric Compton and his three heart transplants. It's an amazing story.