THE EFFECT OF THE SLOWING DOWN OF THE SPIN SPEED, (DUE TO AIR DRAG AND TIP FRICTIONS),
ON THE RISING OF THE TOP
When a top spins and precesses, like in the drawing below, the linear velocity of the top axis, (green line), along the precession trajectory, at the level of the tip, is higher than the linear velocity of the center of mass, (it traces a larger circle with the same angular speed).
In absence of frictions, the two linear velocities would be constant.
In this situation there would be no torque and the top would not rise nor fall.
The tip going faster than the center of mass, (as for linear speed), does not create a torque on the top axis;
an acceleration is needed for to create a torque.

Which is the case of real tops, their velocities are not constant.
Because of air drag and tip frictions, tops slow down while spinning.
The diminishing spin speed of the top causes a braking action at the contact point, so that the linear velocity of the tip along the precession trajectory is coinstrained to diminish, (if the tips doesn't slip, which is the most common case).
While the tip slows down along the precession trajectory, the center of mass would want to continue to go on with its constant linear speed, (inertial speed).
So a torque comes out from this situation.

This torque is in the direction to make the top to tilt more, not to rise.
So, the top slowing down works against the top to rise.
Air drag works against the top to rise.
Rotational sliding friction works against the top to rise too, by both the mechanisms explained, the one of this comment and the one explained in the video.
Rolling resistance is the only mechanism that makes the top rise, as explained in the video, but one effect of rolling resistance is to slow the spin speed of the top, and, in this sense, it fights against the top to rise.
Since real tops do rise if their spin speed is high enough, it can be supposed that the rising torque of the rolling resistance alone is stronger than all the other opposing torques together.