I think if I unsoldered the green wires from the pin and soldered them together, I’d just ground the board while the cage would be left absolutely unshielded.
Yes, but I’m not suggesting you leave it like this.
We want to get to a place where the chassis is grounded (no 33ohm resistor ) but leave the board floating with the 33ohm resistor. This I believe is the optimum solution.
If you can prove the chassis ground can be detached from the board ( by unsoldering the ground post earth and checking that there is no continuity between chassis and circuit board ground ) then you can safely ground the chassis and ground post to the mains ground pin ahead of the 33 ohm resistor with an additional wire. In other words ground the chassis and grounding post without the 33ohm resistor but still run the 33ohm resistor to the circuit board.
You could do as you suggest and just eliminate the 33 ohm resistor and leave everything grounded at 0. Only downside is it may be suboptimal for the circuit itself in terms of noise floor.