There was an old exception for four D3 schools playing hockey in D1. When the NCAA realigned the schools into D1/D2/D3 those 4 schools continued to play in D1 just for hockey (and Hopkins for lax?). Now there is a general rule that a D3 school can ‘play up’ in one men’s and one woman’s sport.
To be in D1, a school has to compete in at least 14 sports, D2 in at least 10, and there are rules about number of women’s, men’s, team sports, seasons, number of games against other D1/D2 teams, etc. Some schools just can’t support that many teams so must stay in D3 or D2. If a D1 school doesn’t want to sponsor a sport, there can be a club team for that sport but not a D2 or D3 (can’t ‘play down’ a division; a few do in women’s hockey, but then they can’t compete for the championship). Texas, an enormous school, only supports 20 teams so you can see how difficult it would be for a small school to support 14 sports.
I don’t blame the men’s rowing powers for not wanting to join the NCAA. Too many rules.