Good advice.remove, clean, replace - if cost effective. some emblems are super expensive (Lamborghini comes to mind at 160 bucks for the rear one). you can remove the emblem in 2 min with heat and fishing line (if adhered) or sometimes by a bolts, or by breaking if snap in style. then hit the whole area with your polishing regimen, then install the new one. Done in 15-30 min in some cases and perfect results...then maintain with weekly washing just like the rest of the car.
I'd follow it up with coating the area with Opti-Coat Pro or CQuartz Finest before installing the emblem again and from there on out cleaning around the lettering will be MUCH easier.