Odds of Getting In
Slim chances. In the first place, there are only 4,000 private golf clubs in the United States (source). The majority of those clubs aren't looking to expand, so they aren't looking for more managers. One manager tends to be enough. In order to be considered for the position, you need to build up a lot of experience. At the very least, we're talking years of running a large business, a hotel, or a restaurant. And it's not easy to become a hotel manager either.
At least your chances of becoming a country club manager are better than your chances of affording club membership once becoming a manager.
Sorry...that was supposed to sound inspiring.