Tonearm for Technics 1200GR

YES.

I think 225-230mm P/S distance is the limited factor in choosing new arm for Technics SL1200GR 

Technics stock arm has 225mm P/S distance. 

So, when looking for a new arm, the p/s should be less than or equal to 225mm. So if the SME is 233.2 mm the the new armboard would need to adjust, is some direction away from the spindle, 8.2mm. 

Is is this generally correct?

You can look at the arm from the top to figure out how much the new arm can be moved within the armboard area. If it’s too far then maybe it will be impossible to use dust cover, counterweight can be out of the Technics plinth. So it depends on each particular arm. Some of them are way different with armtower like Reed or TriPlanar. You can look for the Reed 9’5 (used/demo) maybe ? Great new arm!

Some nice vintage 9’5 arms can be mounted easily, like SONY PUA-7 stand alone version (very nice tonearm). Not cheaper version from Sony turntable, but a version that was sold separately, compare two different version of SONY PUA-7. This arm is fully adjustable, top quality, with the best armlift ever, VTA on the fly. Highly recommended! The price is lower than your limit.

You should first look at adding isolation to the turntable before anything else. That’s the weakness in the SL-1200 series. Your arm is more than up to task. Second area of improvement is in the cartridge. 

I do have an isolation base. And I’m on concrete so that typically helps too.