It's first year kinematics. The (first order) math is pretty easy to work out. 1/2 I w^2 is the rotational potential energy, Assuming all the mass of the wheel is at the rim (worst case) I = mr^2, w=v/r, so that works out to 1/2 mv^2, which is the same as the translational kinetic energy. So the 'energy penalty' of rotating mass is 2x. So, a 10kg bike with 2kg of wheels and 100kg of rider is approximately equal a non-rotating mass of 112kg. However, that rotating mass gives you 2% more kinetic energy to trade for wind drag, gravitational potential, or other friction. So, not much.

A bike's does not require gyroscopic effects to stay up -- there was an experiment with counter rotating wheels (to cancel the gyroscopic moment) that was ridable. The actual stability depends on the geometric and pneumatic trail, flop, weight distribution on the steering axis, and a few other things.

There is a middle ground that people have converged to -- Somewhere in the range of around 600mm bead seat diameter and an inch or two of tire. (26"->700c, 25->50mm width covers most of the biking world.)