First, your assumption number 2 is invalid, as I think you are biased by classical mechanics in your understanding of "point particle" but John Rennie already addressed that point.
I just want to complement on this answer in the classical case where your conclusion would not hold either. For example, Lagrange found in 1772 a family of solution where all three point particles do not move in the same plane. A detailed derivation can be found in Danby (1988). Here is what it looks like, where A, B, and C are the position of the three point particle:
All orbits are ellipses and ABC remains an equilateral triangle throughout.
Many more, way more complex, periodic orbits have been discovered since then. You can find a nice review in Musielak and Quarles (2014). Even if you don't want to or can't follow the math, there are plenty of nice illustrations, see for example figure 3.
So the only question, really, is the stability of such non-coplanar orbits and whether they will eventually become coplanar. Most of the studies of 3-body motion I know of concentrate on a system where one mass is enormously bigger than the others (the Sun, of course). With three equal masses as in your case, I must admit I do not know.
J. M. A. Danby. Fundamentals of celestial mechanics. Willmann-Bell, Richmond, Va., 1988.
Z. E. Musielak, B. Quarles. The three-body problem, Reports on Progress in Physics, Volume 77, Issue 6, article id. 065901 (2014), https://arxiv.org/pdf/1508.02312.pdf