OK, not Thief, but DX and SS definitely qualify as action games. They required the player to use some basic tactics and try to conserve ammo, but if you knew level layout and were good at scoring headshots you could easily massacre small armies. Just yesterday I gunned down the entire Area 51 personnel with great prejudice, using ventilation system to my advantage.
It should, and it's perfectly doable, as Deus Ex demonstrated.