I have a 1960's gas stove (Magic Chef) that won't light. The pilot
lights are fine. Would removing the circular alumium burners and
soaking them in something help?

I also have an air compressor and could blow them out once they were soaked.

Any tips or advice would be appreciated.

Would removing the circular alumium burners and
soaking them in something help?

Probably not. If the burners are in fact aluminum, then the blockage is likely
corrosion, and anything that would dissolve the aluminum corrosion would attack
the metal first. Best to use your air compressor spritzer and any other
mechanical means to open up the gas passages.
The passage to the pilot light is on the side of the burner, and the travel
tube for the gasses is held on with a metal clip. Remove the clip (gently) to
remove the burner assembly and deal with the blockage as above. Any welding
supply store will have torch tip cleaning widgets that have multiple sizes of
cleaning wires, one of which will be just what you need. HTH

Joe