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The pressure the tire exerts on the road is always equal to the pressure it's inflated to. When the vehicle weight increases while tire PSI stays the same, the contact patch (area squished flat against the pavement) increases in size.
Bike tires are narrower than car tires because bikes are much lighter (so the contact patch doesn't need to be as wide), and also because they lean into turns (so the contact patch can't be wide). Bike tires are often larger diameter than car tires because they have more gyroscopic effect and thus make the bike easier to balance. They also make it easier to ride over bumps, but on a road bike (as opposed to a mountain bike) that's probably a relatively minor reason.
I think motorcycle and ebike tires are a little wider (but still round in cross-section, so not like a car tire) for durability reasons because all the forces they're subjected to are larger.
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.
If the vehicle gets heavier, doesn't the tire pressure increase?
No. It often decreases.
Tire pressure is the one main variable that determines pavement wear. It is highly regulated for heavy vehicles. Those have to compensate by using wider tires or more wheels.
Yeah I think in that case they had meant to say that the amount of air in the tire stays the same and the PSI increases when the tire deforms. They are right that the PSI in the tire has to match the pressure on the road, that can only happen if the PSI increases when you increase the load.