The thing is, idealogies and culture AKA influence are intwined. One can't spread culture, their way of life, without simultaneously spreading their ideologies. This makes the whole thing a very natural bond which makes sense.The way I see it, they are both tremendously lacking as separate entities anyway.
Do I want influence to be about idealogy and vice-versa? no
Do I think a steadily increasing mechanic like influence could be used as an additional trigger for idealogical events? absolutely.
The problem with tying Ideology to Influence is that you make Ideology a large-empire characteristic, even more so than it is right now.
That is, doing this means that it's yet another huge bonus for having a larger empire than your opponent. Like we need to stack that deck even more.
Right now, a smaller empire can specialize in producing Ideology and be able to out-compete a larger one on that stat, if the larger one chooses to not to bother with much Ideology. If you change the association as suggested, then the small empire will lose no matter what it does.
Ideology isn't influence in any shape or form. Why would morals be helped in any way merely being more influential? Don't tie them together.
If we're worried about lack of mid/late-game Ideology production (which is an issue), address that directly, not by trying to muck with fundamental components of the game.
My suggestion was to break influence into tiers which increase at an exponential rate, which is decided by the number of planets you control verses the galaxy size. This would be most advantageous to small empires. There are other mechanics they need to balance in a similar manner.
The second part. Idealogy is influence. When your influencing others your spreading your morals. Think Star Wars. The dark side is spreading their influence. Their influence is the dark side. One can't separate the empires ideology and its influence. Sound like a bad example. Its no different then imperial Rome which began spreading christianity, its idealogy, throughout its sphere of influence.