How does precession affect angular momentum?

It does have angular momentum in the vertical direction, so that should resolve your issues. There is a horizontal torque, so the upward-tilted angular momentum vector simply traces out a cone, where it's rate of change is always horizontal. The vertical component will usually be very small, but presents no problems, it just stays constant.

No, you are absolutely right, if your origin is the point $O$ in the figure above, then the only torque is in the tangential direction. There is of course an angular velocity $alpha$ in the $z$ direction. The way it works is the radial component of the angular momentum $vec{L}$ precesses, and $frac{dvec{L}}{dt}=|vec{L}|alpha hat {theta}$, where $hat{theta}$ is the unit vector in the tangential direction. Equating the known torque with $frac{dvec{L}}{dt}$ would now give you the precession frequency $alpha$.

That solution you linked to, probably has notational flaws, because it seems to suggest the radial angular momentum changes through the change $frac{dvec{omega}}{dt}$, which is simply not possible, as there can be no net torque in the radial direction.

The interesting thing is, the gyroscope picks up a small constant angular momentum in the vertical direction as well now, when in the beginning the angular momentum was purely radial. This means the bearings near the point of contact $O$ produce a tangential force and hence a vertical torque, for a very short period initially, to set up the angular velocity $alpha$ of the gyroscope.