Since I'm about to use this material.... I did a little experiment -
The panel I have is "XX" Garolite (like Dave Schmarder uses).... apparently there is a wide variation in finish... I again tried buffing - this time using a medium stiff buffing wheel and a moderately abrasive compound (Tripoli).... After a couple of minutes there was a satiny gloss that will be more than satisfactory for my project....

To the left you see the panel finish as I received it - to the right is the polished section....
BTW -
amyotte - the radio I'm working on is a 3-tube homebrew receiver - nearly identical in design to yours.... it even uses the
exact same 3-circuit tuner and variable cap!!! I'd suspect it was a plan out of some periodical of the day...