Why would the plane keep changing? If there are only these three objects, won't the vectors of their gravitational pull to each other all be on this plane too?

If you define the initial conditions such that their relative velocity is zero or parallel to the plane, yes. But that's not the case in general.

It’s an arbitrary plane, chosen at each moment just so you can flatten it

Not from an external point of view, as you might have a momentum component perpendicular to that plane

(but yes I think you might be right if we're centered on the CG)

One of the necessary conditions here is that the three objects return to their exact initial position, and so does the centre of mass. Initial conditions with non-zero momentum must trivially be ruled out. But this doesn't stop them from having velocities perpendicular to the initial plane that cancel out perfectly, so this doesn't refute the assertion that the planes keep changing.

Sorry, but this is a word salad.

Most of the solutions always have non-zero momentum, including in the initial conditions. And https://numericaltank.sjtu.edu.cn/three-body/three-body.htm includes periodic solutions that move in all three dimensions.

They presumably meant non-zero total momentum. If the total momentum were non-zero, then the center of mass will be moving in a straight line, and will not return to where it began, and therefore the orbit would not be periodic.

Big picture the center of mass momentum is concerned and it does not matter if the system as a whole is moving up or down or to the right or the left. Like the Earth is basically orbiting the sun in an ellipse [1] so far as the sun is concerned and from the viewpoint of the solar system not care so much that it is moving around the galaxy unless we are interested that orbit being perturbed by other stars that we pass near over millions and milions of years.

[1] ignoring the parameters of that ellipse changing slightly and slowly thanks to the other planets

Oh. That makes sense. Yes. The total momentum has to be zero.

All of the things in the solution can constantly have momentum.

Oh sorry, yeah - I wasn't fully awake yet when I wrote it. As a sibling comment mentioned, I was talking about the total momentum. Anyway, it was in response to GP's assertion that GGP may be correct if "we're centered on the CG", which doesn't make sense because the CG can't move.

Each orbit is a spinning top. You pull on a top from the side, it's spin axis precesses.

Oh makes perfect sense, do you have thoughts about real life examples? I did some research using AI and it said there were examples of restricted 3 body problems like the trojan asteroids, but no examples in real life similar to what is in this web app

Put another way, while their positions are one set of 3 points, their momenta are another set of 3 points, and there is no requirement that 6 points will always lay on the same plane.

I wonder what phantom forces would appear when the reference frame changes in some complicated fashion. We get centrifugal "force" when we reconstruct F=dP/dt in a rotating reference frame, what would the 3-body "force" look like?