It's both a mathematical issue (multiple nonlinear equations that relate to each other), and the underlying physical reality as to why that is so. The latter can be seen in when you consider an operational approach to solving the problem: body 1 (say a point source) creates a metric, and body 2 is moving in it. But body 2 also affects the metric, so body 1 gets affected and moves differently and so the metric it creates changes. Tathat in turn affects body 2, and so on ad infinitum. It's their effect on each other and that you cannot simplify it into an equivalent 1 body problem as you can in Netonian gravity because of the very nonlinear effects. You cannot define a center of mass like you can in Newtonian mechanics for that same reason.
So, your point that two produce additional effects is, two point sources, also an issue - as I described above. When you also consider that the two bodies may also have angular momentum it gets worse - the two body Kerr solution is also not known.
What it does not happen in Schwarzschild? You are right, there are still nonlinearities in the multiple equations for even one point source, but for a point source with no angular momentum you have a huge symmetry, spherical symmetry. Also, it is rest frame you consider it at rest, so is is also static. That reduces the set of equations to very few and makes it solvable.
The Kerr solution, for one point source but with angular momentum, took much longer to find. There's still a symmetry but it's axial symmetry rather than spherical, and instead of static and time-inversion symmetric you need to allow it to be time independent but not time-inversion symmetric so the time symmetry is weaker. I forget when it was discovered but it was decades after the Schwarzschild solution was, because the equations were more complex, and the physics had more possibilities with fewer symmetries and rotation.
