This reminds me of film "analysis" that shows how the direct keeps creating wonderful triangles between three points of tension, and our eye naturally finds such triangles pleasing.

Neglecting that any three points make a triangle, so this is basically just saying "a frame with three things in it".

On one level there’s some intellectual wanking about triangles, then as you point out any shot with thee things will have a triangle. I wonder, though, ignoring any attempts to over-analyze, if three is a nice number of things to have in a shot. The viewer can only focus on so many things after all.

Three is also a sort of especially unspecial number. Zero things is sort of special in the sense that there’s no concentration of focus (shot of a landscape that just establishes the environment). If the focus is on one thing, then that’s really drawing a ton of attention to that thing (camera zooms in on the murder weapon and lingers). Two things can often be focused on the contrast between them (the villain towers over the hero). Three is the lowest number that doesn’t have a ton of baggage.

Odd numbers of things in a shot is a rule of thumb in framing.

Interesting! Why’s that?

Well, as with anything related to aesthetics, it looks good.

Even numbers of things have a tension between the center of the group and the center of one of the things.