Correct.
The coax cable has two parts. The "outer" part connects to the screw-on connector and must be grounded. The "inner" part is that copper pin, and must NOT touch ground.
The antenna must only touch the center pin of the coax. Anything else is a short. That SO-239 has some plastic inside of it to isolate the center pin. That's also why there is a plastic shoulder washer between the antenna and mount.
When we throw an SWR meter on it, we'll know right away if it is shorted or not grounded. Short = no SWR at all. No ground = way high SWR. I'd rather have no ground, since a short can kill a radio right quick.
Technically speaking, would the antenna be grounded if you feed that adaptor through the bracket and screw them together?