The pressed-in nut is likely knurled and pressed tightly in place, so might not retain itself later if you were to press it out.
Look at the direction the cable tension applies to it, once it comes out it will probably not hold so well later!

The adjuster itself almost looks like the low-end pot-metal version, installed as a replacement. Verify by removing it and inspecting the threads for a casting seam. These are weak and unsightly as compared to the original part.
You can polish pot metal but it does not sustain a nice finish for long. The good ones are prized little pieces.

Possibly your adjuster is the good chromed steel one but has lost it's chrome?