It’s pretty common to bring up gyroscopes for this when people know a little bit about physics. It’s all over motorcycle forums, for instance.

As you say, it doesn’t work. Experiments have been done where they attach a counter rotating wheel to cancel out the gyroscopic effect, and while it’s a little wonky to ride, it works fine.

IIRC, we’re not 100% sure how bikes work just yet. Every time somebody comes up with a model that seems to be good, someone finds a counterexample that throws it in the bin. Even your explanation of bike trail isn’t all the way there; Razer-type scooters still work without trail on the front wheel.