Does this mean I could in theory steal a lot of influence from other factions by fighting equally for both sides?
In theory, I don't see why you couldn't.
If your faction goes into War/CWar, then fight for both sides and strip the influence from the rest until your ready to clean up and win the War/CWar. The hardest part might will probably be managing to control the influence split between to the 2 factions in War/CWar.
But in theory, if factions in War/CWar are gaining influence from outside of that War/CWar by performing purely CZ work, then yes why not.