Long story short the system isn't working, even after the last two patches including 'fixes'.
It's only been broken for 2+ years, I'm sure they'll get round to it eventually.
@Ur-Quan Lord 13 as an update on my previous post, I looked at the save again, and did a fresh test specifically for ethics attraction (which I shall outline further below). First, about the save - it's 2438, the Cirrulan Coalition (my empire) has 3361 pops, and the expected number of materialist pops is said to be 67% per the tooltip, with the actual number being 46%, again, as per the tooltip. I looked at some individual pops - the synths have an attraction to materialist of some 68%, with cybernetic organics having about 65% - however, the number of cybernetic organics is very low relative to the synths (59 pops). For regular bio pops that do not have "recently conquered" (I just finished eating a Fallen Empire) it's at some 60%, and they contribute 87 pops. Judging by the fact that the expected number of materialists is said to be 67%, not 68%, I'd say the tooltip is actually taking individual pops' attractions into account. Or at least some average of it.
Now, since I don't quite remember how I got the void dwellers in question to this point (it was late in the evening/night), I cannot speak as to wars conducted other than that Fallen Empire war, but I do have a mess of factions, possibly due to being in a federation (with fellow egalitarian materialists I might add). I shall attach the save for anyone's perusal, however be aware that I was playing ironman. (All DLC, no mods)
Now, as to the actual test I did: I started a game as spiritualist/fan. militarist (and swiftly got my posterior handed to me by Commodore AI, but that's another matter). Paying close attention to the ethics distribution, and conducting no diplomacy other than a rivalry that was forced upon me, this is what I observed:
The capital started out with a fairly random distribution of ethics on the pops, as we have come to expect from this new update. By the time factions spawned, there were not enough pops to generate anything other than a spiritualist and a militarist faction (of which the militarists had more support, as you might expect from a fanatic ethic). Now, I did build a temple first thing on my capital, and it did affect the expected pop distribution from 40% to 41.18% or so, and it remained there. I also picked the species trait for +30% gov. ethics attraction, and picked up the One Vision AP for further stacking of government ethics attraction. By 2247, there are exactly 1% egalitarians left out of 90 pops, so it's one pop kicking around somewhere, the rest are as follows:
13% xenophobe / 26% expected (I'm in a defensive war against xenos right now, the second one. They started out at 10% or so when the game started)
47% militarist / 34% expected (Bit surprising that xenophobia is so relatively strong, but then again, I don't blame my pops, brutal wars being lost)
37% spiritualist / 40% expected
I will note that I used the isolationist diplomatic stance for a while before coming into serious contact with other nearby empires to further stack unity and ethics attraction.
Now you might say that those numbers don't quite add up to 100% and you'd be right, I expect that particular tooltip not showing decimals is the culprit, seeing how I don't have exactly 100 pops. However, what this test showed is that pops do shift ethics. Quite rapidly, in fact. My capital only has that one egalitarian pop left, all the others are as you'd expect. In fact, the ethics shift fast enough for a defensive war to cause a xenophobe faction to spawn and grow.
I will note that I did not use any promotion/suppression of factions - as you can see from the numbers and context provided, outside factors influence internal factions quite heavily (as does the ruler's faction in an oligarchy, who'd have thought). So, to conclude - pops shift ethics, and they shift them in the general direction of what you'd expect, however accessing the breakdown of factors and their actual attractions is ... a thankless task. Moreover, due to the arcane nature of attraction modifiers, it is often difficult to understand exactly what contributes how much to the overall ethics attraction.
Right, I promised save files, so here they go. Ironman save is the materialist one, the other one is the test save, if anyone wishes to try to rescue what I view as a hopeless cause. For anyone wanting to take a look, I'll reiterate - saves are using all DLC, no mods.