Height does effect TAO, no debate there.
Ground radials? The other half of the antenna per say.
Many verticals, ham antenna's included use the coax shield as the counterpoise, the coax radiates just as much if not more than the antenna itself. It is designed that way from the manufacture.
A no ground radial antenna is half the antenna.
Using ground radials on elevated verticals, in some situations help prevent CMC, Imax 2000 etc type antennas. Do they increase the performance? Maybe, by forcing the current onto the ground radials instead of the coax.
The current then becomes symmetrical per say and this results in a cleaner pattern of radiated signal.
Some will argue that their vertical with no ground plane out performs a vertical with a ground plane. I always ask "In what direction?". As a no ground plane vertical is left to radiate in a direction effected by the surrounding soil conditions, type of install and many other variables.
A vertical ground plane radiates poorly in all directions if properly installed.
But they do radiate.
