If you're saying that a prominent component of the motion of a stiff unbalanced top is fundamentally non-gyroscopic in nature, I totally agree. In my experience, this component has all the characteristics of rigid-body whirl. The "loose bearing" part is critical to the extension of whirl to unbalanced tops.
I believe that there is not gyroscopic effect related to the component "unbalance wobbling".
There is gyroscopic effect in a spinning unbalanced top, but it is related to some other component of the whole motion, which is the component "precession", and/or the component "nutation". If precession and nutation are absent, still there should be a sort of "micro-precession", (I don't know how to call it), which keeps the axis of rotation in vertical position, so, at least a very tiny amount of gyroscopic effect is always present in the whole motion.
I think you are into something, when you say "whirling".
I remember when Aerobie described one of his tops starting to wobble spontaneously at a certain speed, and then ceasing to wobble at a lower speed. I saw the same in my tops, (when the tip is very weared).
This wobbling is nutation, (at least that I saw in my tops); I can obtain nutation kicking the stem of the top while it is spinning, but in these cases the nutation started by itself, after minutes of the top spinning in sleeping position.
I suspect that whirling could have something to do with all this, triggering nutation through the tip, (in some way a bearing, as you say), and only at a certain speed, because of a resonance effect.
But unbalance seems something different to me.
When you say that the tip scrubs in an unbalanced top, (I see the same in my unbalanced tops), my explanation for this is that the center of mass is not vertically aligned with the tip, (which is what it makes the top unbalanced), so, especially when the top spins fast, the center of mass wanting to stay in the axis of rotation pushes the tip out of the axis of rotation, (since they are not aligned), and for this reason it scrubs on the base.
The component "unbalance wobbling" is the simplest of the components of motion in a spinning top, because it doesn't involve a gyroscopic effect;
think to a perfectly balanced top (with the tip in the axis having the largest, or littlest, moment of inertia, passing through the center of mass, which is what it makes a top balanced) but a distorted shape and a bent stem.
Even in sleeping position, such a top would still seem to precess, but it would not be a real precession, it's only a visual effect.
It's not very different in the unbalance motion: it's not a real precession (involving gyroscopic effect), but just a visual effect.