Heh this isn't wrong after all. Though it would be weird if it were. Ofcourse an individual is always at a disadvantage compared to a group since a group has more resources in manpower. But it doesn't make a difference if you play a lone wolf in a different mode. Your impact will be of the same effect and scale.Solo players are generally at a disadvantage against Open players in BGS conflicts. The reason for this is that the Solo player is more likely to be a "lone wolf" just trying to do his own thing, whereas Open players are more likely to be part of a large group. As previously mentioned, manipulating the BGS is an exercise in competitive bucket-filling, where the advantage goes to the larger group. A lone-wolf needs to struggle VERY hard to compete with that.
Victory should go to whichever side can fill its buckets better, whether through superior numbers, or superior determination. If a group in Open can't do that, they deserve to lose.
However, "filling buckets" is merely the basic task there is to do. When a player is doing it for itself, sure, the bucket filling task is the first and last thing to do. However, when player founded BGS factions are competing against eachother, racing to fill buckets is a boring and imo pathetic way of competition. After all, one possible task to fill your own bucket faster is to slow down the progress of your opponent filling theirs. So if you can deny your opponent(s) progress or even revert it and cause a setback, you will be at a serious advantage, however, the same effect can happen to you too. The result is a much more dynamic system where results are uncertain until the end of the BGS cycle. Whereas filling buckets is, as you portrayed, merely a boring bucket filling tasks where the larges group wins (due to the fact that the bucket filling activity is not challenging and includes next to zero chance of failure). The outcome can be easily calculated by considering the amount of players and time each party has available.
The result: a linear, indirect competition of two player groups that don't even know what the other is doing and why.
The OP suggest that (for player groups and player competition), the influence game should follow uniform rules where any kind of gameplay the game has to offer can be used to a group's (dis-)advantage instead of a game that has different rulesets for each participant.
For NPC factions only I too do not see a reason to change the influence game as, after all, it is a background simulation. Excluding the others modes will most likely result in less accurate data. But moving player founded BGS factions to open only is IMO a change that is a requiernment for dynamic gameplay.