Some, like the Parkers are set in place with multiple tiny nails made of the same material as the oval. All in place around the edge of the oval into shallow countersinks and polished off flush along with the oval and surface of the wood during finishing.
Some are soldered blindly to a wood screw that is first set into the inletted area and carefully adjusted to depth and it's head filed (generally) so it lies just under the back of the oval when placed in position. The screw head is them 'tinned' with soft solder, the back of the oval is fluxed and layed in place. The two are pressed into place and held w/a soldering copper as the solder between the two melts and joins them at the back. The outer surface then gets a finish polish to touch up and define the metal to wood fit.
New work is generally just epoxied in place now.
You can see to make it really fit perfectly, it takes some polishing of the fitted oval to the stock surface around it. On a finished stock that also means some wood finishing beyound rubbing some magic linseed around the area to make it look right.
On a oval still in the wood that you try and remove the existing engraving w/o removing the oval itself,,you run have a couple of things against you.
The biggest thing is that you have no idea how thick the oval material is, and how thin it will be when you get through polishing out the existing engraving. It may look great, or at least presentable, but when the engraver touches the surface to cut the new work, it may indent or even break through.
On thing for sure, you don't know till you try!
Also with this attempt is it is very difficult to polish out the oval w/o scarfing up the surrounding wood and you have the same wood refinish problem as above.
Just some thoughts.