3 faction sometimes works, sometimes does not. Let’s take your example and merge it with two MMO examples.
Planetside – 3 factions. I absolutely hated three factions in that game. Seriously. There was nothing worse than spending an evening hammering it out with the TR (I was NC) only to have the VS come waltzing in to back-hack us with minimal interference. I know this is supposed to be “two sides teaming up against a larger third” but when the VS had most of the continents locked and we were having a hell of a battle without them it was just a sour note that screwed over an entire evening’s gaming.
Battlefield: Europe (WWIIOL) – 3 factions, 2 sides. Brits/French on one side, Germans on the other. Far better of the two games precisely because there was no third faction to come strolling into the backfield taking towns without opposition. In fact they put a nix to that happening early on through the use of links, offense and defensive links. Something Planetside also did as well.
3 factions is a holy grail to some people. I, however, remain unconvinced that it is the holy grail for all people. It certainly isn’t for me. I only see it as a mechanic by which two sides are in for an unfulfilling night.
1: The faction that loses territory to another unopposed.
2: The faction that gains territory by sitting around, unopposed.