Angular momentum and gyroscopic precession are critical to keeping a bicycle balanced. To see why this is the case, see what happens if the brakes lock the wheels in place while the bicycle continues to move forward by skidding along the road.
It sounds like the problem was redefined in such a way as to eliminate the 'conservation of angular momentum' as being a significant contributor by someone who invented a very awkward bike that remains stable without angular momentum. If you are talking about a normal bike that people actually ride, then it is you who is redefining the problem. Perhaps normal bikes do rely heavily upon conservation of angular momentum in order to remain stable. The fact that another bicycle was created that does not rely on this effect says nothing to those bikes which do. In other words, it is a false equivalence. The term bicycle is much too vague, as multiple types can be constructed that rely on different factors in order to achieve stability. Just because one type of bicycle eliminates the need for one factor does not make that factor irrelevant for other types of bicycles.
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