Not true. Try playing a disc priest, a holy paladin, a resto shaman (this not so much right now because due to their state you could probably click everything and win anyway), a warlock, etc with just 5-7 keybinds. You won't be able to, or you will, but you won't ever get above 1800 rating. Personally I have around 30-50 keybinds for every character I play, but healers take the cake by a longshot (only DPS that comes close is warlock, hence I mentioned them specifically).
As for the OP and if he's talking about arenas and PvP in general - Frost DK. Necrotic Strike, Frost Strike, Howling Blast, Grip, Hungering Cold, chains of ice, death strike, IBF, AMS, PvP trinket (and possibly an on-use trinket or a racial), Lichborn, Deathcoil, Pillar of Frost, Dark Simulacrum, Pet summon + sac and... Yeah, that's pretty much it - 15~ keybinds and you should be good to go. There's obviously more shit you'll be keybinding as you get higher.