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This is merely a convenient approximation for properly-inflated tires carrying a load, not a hard rule rooted borne out during empirical examination. After all, removing a wheel from an automobile and rolling it along clean concrete leaves tire tracks that are full width, yet the tire will not substantially deform at the contact point because 20-30 pounds is not much of a burden. If there's no deformation, then the contact patch is a line with a tiny area, which would wrongly suggest a ludicrously high tire pressure.
While bike wheels do act as gyroscopes -- as do all rotating masses without a contra-rotating mass -- this is not substantial to bicycle stability. If it were, kick scooters or e-scooters which have substantially smaller wheels but with the same physics as bicycles would be unrideable.
The bicycle has existed for about 200 years, and for most of that time, how it remains stable was an open question in physics until roughly the late 20th Century, when researchers built enough intentionally-bad bicycles to prove what was minimum and sufficient to have a functioning bicycle. This empirically ruled out trail, caster, and gyroscopes as necessary factors. But the most prominent factor that remained necessarily is centrifugal balancing, aka leaning/banking. Turns out, bicycles lean into curves just like airplanes so.
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.