I don't mind playing any role, and have raided as all roles quite a bit. The main reason I gravitate towards healer or DPS is simple: you need a lot more tanks in dungeons than you do in raids. If you want to have a spot in raids, it is a far better choice to be DPS or healer.
Solution to this would be to allow for more tanks in raids. Make it less about them taking a lot less damage and healing themselves and more about taking a bit less damage, healing or otherwise giving support to the group, and dealing a bit less damage than DPS (not the 50% or less they do atm), that way you'd easily get away with having 3-4 tanks in a raid (as the 1-2 extra tanks would make up for having one fewer healer and being able to use another DPS). That way there'd be a balance between tanks and healers in raids and dungeons, and you'd see the tank numbers increase.
TL;DR: Turn tanks into bruisers. This would also solve their PvP issues.