I don't think so. Transfer rates amongs all student has been consistent averaging about 33%
More Than a Third of College Students Transfer | Inside Higher Ed
The transfer rates among athletes is significantly below that of regular students-Men's basketball is the worse.
Tracking transfer in Division I Men’s Basketball
Somehow these academic institutions manage what you are describing as potential "unmitigated disaster".
Quality control, common sense, and much more careful consideration (players and coaches) would prevent much of what you are suggesting as issues. Unscrupulous programs are going to bend what ever rules are in place. The transfer rates might actually go down. For example, I would like to see the foolish coach who would agree to take a player when that would be the player 3rd or 4th school. The scholarship limits and years to complete eligibility are still in place so the numbers of opportunities to transfer are exactly the same as today, so why would the transfer rates quadruple? Perhaps you did not realize that a transferred student's scholarship counts against the limit in the year the player is sitting out.
Lastly, and I should have been more clear about this in the original post. For non-revenue generating sport which in includes WCBB there is no "monetary" advantage for a group of students transferring to one school. There is also nothing preventing a group of athletes from agreeing to go to the same school, except the scholar limit. Years ago the Fab Five did it at Michigan and Diamond Deshields & crew did it in 2014 at UNC.