We have no observed examples in nature of three body equilibrium. But then again, all places we have looked are either influenced by the chaotic orbits around them of the Solar System, our surrounding galaxy, or nearby galaxies in a cluster.
There aren't a lot of orbiting three bodies without external gravitational influences disturbing them.
This is of course a relative statement. Every object affects every other object, subject to the limitations of lightspeed propagation of gravity waves through expanding space.
But as you point out, we still haven't noticed any examples that are stable short-term.
(of course in real life your plane would keep changing, and probably some other complicated math I can't think right now)
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?
(but yes I think you might be right if we're centered on the CG)
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.
[1] ignoring the parameters of that ellipse changing slightly and slowly thanks to the other planets
All of the things in the solution can constantly have momentum.