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Sure, the tire itself has a certain amount of strength, but (unless it's a run-flat tire, I suppose) it's negligible compared to the load carrying provided by the tire pressure.
No, you're overstating your case. First of all, I didn't say that gyroscope forces were the only factor. Second, they are a "substantial" contributing factor. Your own wiki link agrees with me:
The important part is the "gyroscopic effect... contribute" part, not the "solely responsible... discredited" part.
Remember, OP's question was "why are the wheels big," not "why do bicycles stay upright," so the effect that's relevant to discuss is the one that's different between wheels of different diameter. And that's the gyroscopic effect, not any of the other things that contribute to bicycle stability but don't depend on wheel size. There's a reason people generally don't prefer things like Bromptons unless they really need the packaging advantages, and it's because bikes with small wheels are (relatively) weird and twitchy to ride.
Gyroscopic effect for bicycles is neither significant nor necessary. How are bikes with 12" wheels or less going to take advantage of that? There are some functioning bikes on this page whose gyroscopic force would be less than 1% of the mass of the bike and rider. They're certainly a contributing factor, to varying degrees, but even on bigger bikes they aren't substantial. Some guys at Cambridge went out of their way to prove that.
My comment was in reply to the "always equal" assertion, which it definitely is not. No doubt, it's a handy rule of thumb but nobody should walk away thinking it is a hard rule of tire physics.
Correlation does not prove causation. You assert that bicycle wheels are big because they have more gyroscopic effects. That is a correlation. I assert in my other comment that small wheels would be swallowed by potholes. That is a causal relationship: the wheel must be bigger to deal with real roads AND is something a smaller wheel cannot handle. It is a fact that a big wheel rolls over protrusions and holes that a small wheel would fall into.