Honestly I don't like this idea at all. I actually think your current method is pretty solid, though obviously needs polish and balancing. But the biggest advantage to this system is..... it finally divorces culture from expansion.
Expansion already has tons of benefits, and its hard to play tall. Culture at least gave some parity between the two styles. If you make it all based on influence, well then the expansion players gain the benefit again, and it means the scaling of culture becomes a much harder thing to manage.
If I was going to pick any stat to base off of influence.... its control. Its easy right now to burn up your control pretty quickly, and so being able to generate it through "cultural control" and use it more often I think is a fine idea.