Indeed, the internal PSI of a tire is not the only force at play on the contact area of a tire. Contact patch size of a tire made of a balloon would alternatively be based almost entirely on psi (think
rolligon as an extreme example of low pressure, high contact area tires).
I think you're trying to say that some vehicle weight is transferred circumferentially through the tread too, and not only vertically through the sidewalls.
You are very correct, and your point can be made by looking at the
Michelin Tweel. The Tweel has zero sidewall and zero psi:
I actually sat through a lecture a few years ago about how the Tweel operated. The interesting take away was that the internal rubber spokes actually took very little compressive loading... as you would think it did. The stresses are actually transferred mainly through circumferential compressive loading of the tread, and the wheel actually HANGS off of the internal spokes, loading the spokes above the wheel in tension more than the spokes that are in compression above the contact patch. Think of a piece of paper. You can apply a tension load easily, but a compressive load will just cause the paper to buckle.
Sidewalls act similarly. Much of the tracking and side loading strength of a tire is in the construction of the tread. The sidewalls don't provide much vertical compressive support because they are actually in a buckling condition, not so much in a cantilever beam condition. The sidewall above the wheel actually carries most of the carcass load in tension (I know, kinda counter-intuitive) and transfers it to the bead, which acts in axial tension holding on to the wheel itself.