The main point is that you can play 2vs2 even if your partner is not online.
We could combine these ideas: before you start searching a match you can select whether you want to play with your partner or with anybody.
This way we could have all players interested in 2v2 using this system. The only drawback is that complete pairs could refuse to play and still not get penalized (because it would be 2 words against 2 words). We could however ban those players from using the system if we detect such behavior.