This idea reminds me of a league I once proposed where each player submitted a "poison pill" type player, and then each of their divisional opponents had to include the player on the team. This meant each team had five "poison pill" players, but it avoided the problems with random assignment (since only division assignment would be random) and with a draft (since no one would get an advantage by picking earlier). If I remember correctly, I proposed a higher cap, mainly so that the teams weren't forced to use only "poison pill" players, but it low enough such that they couldn't avoid playing at least some of them.
The league never got going, but I still think it would be very interesting. This is close, but not quite the same. (The element of sticking your opponents with a player is fairly diluted here.)