I don't really get the point of your post other than "lol TL;DR."
However, I'll take the opportunity to point out that the average rapid rating continues to decrease each day. It was 867.02 when I wrote my previous comment, and now it's 848.26.
Current average chess rating
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>The average was at 1200 and it goes down due to news players starting with 800.
I've been tracking the average rapid rating every day since 10|0 was recategorized as rapid last September. The average rose to ~1025 just after the change (when the site was setting active 10|0 players' rapid ratings = their blitz ratings) and has been decreasing every day since then. Today it stands at 867.02.
I think that you're correct in your explanation of why this is happening. The site allows players to choose their own rating during account creation based on how they answer the "what is your level of chess experience?" question. "New to chess" = 400, "Beginner" = 800, "Intermediate" = 1200 and so on. Couple that with millions of players having joined the site recently due to The Queen's Gambit, and most likely having chosen starting ratings of 400 or 800, and it's not surprising that the average rating is decreasing. Based on the changes I've observed in the population numbers on the leaderboard (and that used to be shown on the old stats page), these new players represent a substantial fraction of the total rapid pool.
My understanding of how the rating math works is just as you say: the average will be more or less equal to the starting rating, and the strength of the new players won't change this. For example, if the average were 1200, every player started at 1200, and a massive influx of weaker players were to join, then we'd expect the average to remain 1200 but for ratings to inflate so that the new 1200 represents a weaker level of strength than the old 1200. I don't think that we'd expect the continuous leftward shift of the peak of the rating distribution that we've been seeing.
There's a sense in which all of this doesn't matter because rating numbers are arbitrary, but there's also "common knowledge" in the community regarding what level of strength a given rating on the site represents, and I think that these two factors of "choose your own rating" and millions of new players joining are shifting the underlying reality a lot faster than people's perceptions of this common knowledge are changing, leading to people wondering and complaining about "how is this kind of play happening at 1000?" and similar observations.
For this reason, while I understand the choice to allow "choose your own rating" to improve new player experience, I think it might be doing more harm than good relative to the old "everyone starts at 1200 without exception" approach. If things continue on the current trajectory, I can see average ratings getting pretty low depending on just how many people are choosing "new to chess" (400) vs. "beginner" (800).