Lot of interesting responses here but it has nothing with steering or gyros, its just angular momentum.
When the bike is going forwards, it’s vector of momentum is straight forward matching it’s vector of movement, meaning that no net acceleration is being applied laterally to it. You can steer how you like.
When the bike starts to tip over, it naturally tries to move to steer to one side or the other, that’s why you can steer by just leaning if you want.
While you are leaning/turning, the vector of movement is straight forward where the bike is now pointing, but the vector of momentum is still pointing near where the bike was pointing before. This means that there’s a net force on the bike sideways pushing it back upwards. This is why it’s naturally correcting.
Next time you are riding your bike, try steering by just leaning and notice how you don’t fall over, and the bike will self correct. Then lean over while stationary and watch yourself fall over :p
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