Units of angular velocity and frequency

Question

I was recently studying for an exam and had a small argument with my teacher when I said that the angular velocity of something was in hertz. The way I see it: $dim(ω)=\frac{rad}{s}$ but radians are by definition dimensionless (in the original definition a radian is a metre per metre from what I know), so the units come to $dim(ω)=\frac{1}{s}=Hz$.
He said that maybe the units cancel out to hertz but that the meaning of hertz is an occurrence per second and it does not describe angular velocity the best.
By that logic we should measure frequency of rotations of a system in rotations per second but we measure it in hertz.
In addition, from the formula $ω=2πf$ we can see that angular velocity and frequency should have the same units.

If we measure frequency in hertz and the meaning of it is
"how many times a system completes a cycle per second"
or in different words:
"how many times a system completes a rotation of 2π radians per second"
I'd say that measuring angular velocity in hertz would mean:
"how many times a system completes a rotation of x radians per second"