I think this is a good idea. It simplifies things for the purposes of record-keeping and makes pre-election vote counting easier for people who are trying to push through (or block) legislation. At the same time, I didn't feel like the stat-based influence really had any impact on role-playing, which is the most important part of the game.
What DID have an impact was influence earned by player actions. This was most notable in the bonuses for the Ex-Megas and for people who managed to marry a Princess. I think that it would work well to give everyone a base vote of 1, and only allow additional influence beyond that due to similar in-game actions. Other options along these lines that could be explored:
1) Influence granted by the consent of the Senate, such as a 'triumph' for a great military victory or some other significant achievement.
2) Influence granted to someone who has won a PvP battle.
I also think your truncated rank system is a big improvement. LotR ended up having too many ranks and it was too difficult to reach the highest ranks. Reaching a high rank should be hard, but not impossible which is how it turned out in LotR. I think your system will work better.
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