Unless I was already seriously considering joining another club, I'd vote yes pretty quickly.
I assume I've already paid a $25,000-$50,000 non-refundable initiation fee, so I don't want to walk away from that and have to pony up another one to join someplace else. And since I live in metro NY, access is a major problem so I want to be in a club.
Most importantly, the course is pretty useless with a "toxic" condition, so I would feel a sense of obligation to my fellow members not to cut and run, leaving the financial burden on a smaller number of members.
However, I'd make sure that we explored the possiblity of having a better course after the work. Maybe add more contour if the greens were boring, or restore lost features, certainly improve drainage if that was a problem, etc. Although it is a major imterruption, having 10 months to work on all the greens is a rare opportunity that many courses cannot consider.