Just looking at a straight calculation of numbers of factions and rules perhaps doesn’t give a full picture of the difficulty of balancing game rules.  The relationship between rules as several above have mentioned, is probably the key area that takes the balancing problem from very difficult but still possible to model numerically, to nigh-on impossible.

Firstly there are game interactions that cause synergies (where two rules combine together to create a greater effect i.e. 1+1=3).  Then there are more complex interactions as @lawnor mentions putting in higher complexity. Some games offer options to daisy chain rules and effects together to create bigger effects (Guildball is a classic for this) and then offer options like damage selection so that the end effect can be different based on gamer choice. That means that rule effect +rule effect = a big number of potential outcomes, not just one. For example, a player can choose a particular effect from one unit, combine it with one from another unit and choose a range of options that result in damage which can be applied in a wide range of ways to hurt the opposition and/or gain improvements like momentum that allow different rules and effects to be used in further combination. It is hard to make a model that can show this numerically so this would suggest that rule balancing is always going to be fraught.
On the same topic perhaps complexity in rule use and combination isn’t the biggest problem. Lots of gamers value complexity as it can present tactical advantages (Alpha strike anyone?). Therefore, the main concern is dysergy or the opposite of synergy. This represents for me the negative interaction between rules (often new or changed rules) which bring disharmony and “breaks” the game. Game design and redesign are both very complex and any attempt will have both intended and unintended consequences.  The key thing to avoid is breaking the players’ gaming experience.