A circular orbit is when a linear force is applied orthogonal to linear velocity. The direction changes, but not magnitude (speed).

Gyroscopic precession is when a rotational force (torque) is applied orthogonal to rotational velocity. The axis of rotation changes but not magnitude (rotational speed).

I've never thought much of it.

(This is not a comment on the precession video itself, which haven’t clicked on)

I’ll look up the Dennett article, thanks.

I don't see how this is not magical thinking but the rules of arithmetic are.

I don’t recall any “magic thinking” in the explanation of precession in my introductory undergraduate physics text. Just Newton’s laws and vector cross-products.

“you are applying the lateral force to a different spot on the object at each time interval” : how is that applied when the force is gravity, applied to every point of the object at every time?

It's about the point of suspension resisting gravity. The axis is "held" at a point, applying an "upward" force (resisting gravity). With gravity as you say evenly acting, the point opposite that through the centre of gravity will have a downward vector. The natural result of these opposing vectors might be rotation through the centre of gravity, but when those rotational forces are applied to moving targets you get suspension from the product.