I don't think your question is entirely clear. Are you asking whether the physics on some other plane through the origin might be different than the physics on the theta=pi/2 plane? That would contradict spherical symmetry. In essense, the stipulation of spherical symmetry is tantamount to the stipulation that all planes (and rays) through the origin have equivalent physics. So your question is "can anyone prove that a spherically symmetrical field is spherically symmetrical?".
One could question whether the gravitational field of a spherical mass (or a hypothetical mass point) is really spherically symmetrical, but by the principle of sufficient reason it would be hard to justify any such hypothesis without also positing some lack of homogeneity and isotropy in the universe. But this probably isn't what you are asking, because you asked for a "proof", whereas this would be an empirical question.
By the way, I was able to view 177 and 178, but I don't think that discussion addresses the question, because it assumes that a spherically symmetrical field is spherically symmetrical. It doesn't present any "justification" of this, it treats it as tautological, which it is.
I suppose one other possible interpretation of the question is: How do we know that a test particle will move entirely in a single plane. Needless to say, this isn't related to the theta=pi/2 plane, and obviously a test particle need not remain in that particular plane, but one could ask whether a test particle's motion might not be confined to a single plane. This too is rather tautological, since the trajectory at any point is in some plane, and there is no momentum outside that plane, so what would cause it to diverge from the plane? And in which direction would it diverge, since by symmetry it would have no more reason to go one way as the other. If this is what you're asking, then the "proof" you're looking for is simply "conservation of angular momentum".
