If you made a bike with a vertical head tube or negative head tube angle, it would not be stable at all. Gyroscope effects are minor in the mechanics of bicycles.

https://road.cc/content/feature/bike-geometry-101-learn-why-frame-angles-trail-matter-267512

There's a few geometric parameters that influence straight line tracking. Try holding a bike by the head tube and move it very slowly forward. Gyroscopic effects are negligible at such speeds. Yet, you should notice that if you lean the bike left, the front wheel turns left and vice versa. This is due to the geometry of how the front tire pivots. Since the front wheel turns in the direction that the frame leans, as the bike rolls forward, the centerline of the tire contact patches moves in the direction of the lean restoring the frame to vertical. The reason it is more stable and smooth at high speeds is that the contact point of the front tire moves faster with speed. That makes for smaller deviations from vertical.

A big part of learning to ride a bike is actually learning to countersteer before turns. Suppose you want to turn right. If you start the turn by turning your handlebars to the right, the wheelbase will move to the right and the frame will lean left. Without a quick correction, the rider will topple over the left of the bike. New riders have to train their brains to instead initially turn a little to the left to establish a rightward lean before they can start the turn. Then at the end of the turn, the rider oversteers to get the wheels back under their center of mass before pointing the wheel straight again. This can be easily seen if you ride on a sandy gravel. Make a few turns then look at the track that the tires make.

All this is to say that the biggest factor in stability is that a lean naturally turns the front wheel toward the direction of lean. Traveling forward moves the tires back under the rider restoring a vertical alignment.