Tim, you asked a fair question, and I'll give you an honest answer...
The only circumstances I'd accept an invitation to ANGC is on the barter or exchange basis whereby (if they still have this tradition and policy) those that volunteer to work at the Masters in key roles like extra grounds crew for the event, have in the past been allowed to play the course on one of the last days it is open for the year going into the summer layoff-closedown. I would quite enjoy a job on the grounds during the toon-a-mint, and then play with other similar volunteers.
I don't live in the members world mostly. I'm sure there are a lot of swell guys there that once you cut through all the up-front exclusivity and pretense, and are just regular Joes that love their course. But, I don't feel comfortable running in that pack... So, if I could play on a day when the hoi polloi are having at it, I'd be thrilled.