These tires were a special heavy construction by Michelin, which if I recall correctly Budd obtained under license. (There was an article on the Silver Slipper in Trains in the early '70s that covered some of the details but did not go heavy on the technology). One note was the reported high number of 'flats' that occurred on typical jointed rail; this led me to think the 'system' involved relatively high-sidewall tires kept aligned by the inner steel flanges, but that would not be correct. And in fact some of this might involve other problems than just 'no air'...
Something interesting that came out of the '70s that would curl hair over on the main Trains forums was that the Chevy Nova family somewhat conveniently had a front track that matched standard gauge almost if not exactly. Railfans discovered that if you drove one of those cars up onto the rails there was a strong 'self-steering' effect, especially if you let a little air out of the tires, that would let you go considerable distances without touching the steering wheel or 'hitting the ties' even at some speed. The real reason this worked, I think largely unrecognized at the time, is akin to the patented "mechanism of action" in the Michelin system: the tread face of the tire is negatively flexed until concave over the railhead, and this explains why there is rudimentary if any visible tread: any lateral perturbation or curve force results in the tire 'self-centering' even if there is considerable bounce.
There are obvious consequences if you get any reflected energy that shows up as vertical 'bounce' -- yes, there is a metal flange in case the self-steering force is exceeded, but you can bet if there is any reflected yaw building up (an additional likely reason for the five axles plus pony guides) there will be interesting unanticipated vibration modes in the suspension...
But more importantly, this is early-Thirties rubber and carcass design, with the tread, sidewalls, and shoulder all being repeatedly and severely flexed every time the contact patch is loaded. No surprise you get seps, cracking, etc. -- or that the tire comes apart dramatically well when the air pressure goes out and the tire gets pushed down toward internal rim contact (you can see the evolution of this in the RATP system with the whole steel wheel and rail arrangement inboard of the tires, and flat running surface contact of a normally 'domed' contact patch)
Now load it heavily enough to keep the ride from bouncing like a basketball, in that age of Houdaille or friction dampers ... including lateral component driven by the elastic rebound of the relatively high pressure. And remember also what happens, especially acoustically, when the ride runs out of compliance either against the guides at the ends of the trucks or the flanges.
After all this, we get to the issue of braking in the presence of water on the rails, let alone organic matter of the general sort that poses adhesion issues. There is a reason "high traction" asphalt composite using ground-up tires to enhance braking ... something that can be demonstrated to make substantial stopping-distance reductions on test ... never caught on, and the answer is succinctly 'moisture' -- when it's present, rubber-to-rubber contact turns into Slip City.
Looks to me as if the rubber donuts were derived from inner-tubes, probably truck inner tubes, but I don't think that's a vestibule door to go from car to car (and just look at the thin gauge of the sheet metal used in the bodies!!!) -- I think it's a means of reducing perceived surge between tender and train when a steam locomotive with longitudinal augment is being run in one of its critical speed ranges. The air cushion has no inherent 'spring rate' or harmonic as something like a Franklin radial buffer would.