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So, yes, use a building slot for a holotheater and save yourself 8 pops.
If you don't care that much, yes, add Holo Theatres when you run out of Amenities.
I'm unsure if population happiness effects faction approval but I know faction approval boosts or reduced population happiness.
If your Stability is good enough, then you don't have to worry about Amenities, with the caveat that my notion of "good enough Stability" is quite a high one, and I genuinely don't have any experience running low-Stability planets. So I know little about Revolt chances and so forth.
The thing to note, here, is that low Amenities have a bigger and bigger impact on Happiness (which in turn impacts Stability) for each 1 you're missing.
At 0 there is no effect.
At -1 there is a very, very tiny effect.
At -2 there is a tiny effect.
At -3 as amall effect.
At -4 a somewhat bigger effect, and -5 a bigger one still.
It is absolutely *not* linear. You'll probably be very fine running at -1 or -2 Amenities for quite a while, maybe even -3, but look at that which truly matters: Stability.
Ignoring Gene Clinics (and Ruler Jobs, which are basically hard to get more of), Amenities come in chunks of 20 and can at most be halved into +10 Amenities from one Entertainer Job, if you close the other Job slot from the Holo-Threatre.
I've played very little v3.4 (waiting for the v3.4.4 bugfix patch) but while I played what little I have, I noticed that Clerk Job assignment seemed much better than in v3.3 or many previous versions. POPs will much more strongly prefer other Worker Stratum Jobs unless there is a need for Amenities, in which case they'll fill some of the Clerk Jobs.
I've played little, I want to emphasize, but based on that it seems to me that one could perhaps leave a few Clerk Job slots open and rely on the system automatically shifting Workers from other Jobs to Clerk Jobs if the Amenity situation becomes bad but not so bad that it warrants another Entertainer Job, and then shifting those Workers back to their regular Jobs when the Amenities from their Clerk Jobs are no longer needed.