Dang it Despotus, you said everything I wanted to say!

Well, except that it's easier to breed one monster type at a time. Some people feel bad about ditching Boco, but he is a generic chocobo, and you can always rename your generic monsters. Personally once I've bred monsters of every class that can net rare items I recruit a Hyrudra in Nelveska Temple and breed up two Tiamats. They hold a position of honor in my roster in every game.