Thomas,
I have long ago disassociated myself from such delusions. I wasn't even thinking about my son in Manhattan because, unfortunately, his work and place of residence only allow him some 8 or 10 annual rounds.
But back in the glorious days of my youth, I might have considered such a challenge. Coming from modest means, I have used similar opportunities to play courses not otherwise available to me.
Occasionally, like in the Columbus District Golf Association, I accidentally qualified for the District Amateur and got into the Open (Columbus CC, Muirfield Village, respectively). Typically, however, I picked the best qualifying courses to play, not necessarily those where I would have a better chance of qualifying for the tournament. The serious tournament players always did the opposite, qualifying at their home courses or at others they were very familiar with.
By the way, the CDGA requirements, as I recall, included membership at a member club (the Scarlet and Grey Golf Club was mine at a student cost of $100 per year), and a handicap of 4 or 5 for the Am, and 2 or 3 for the Open. The Texas Golf Association had similar membership and handicap requirements, though the level of play was substantially higher.