[Arran]

Grounding the chassis did the trick The interference isn't completely gone, but it's greatly reduced and no worse than any of my other radios.

Now that I think about it that makes sense since both line capacitors and all of the coil and tube shields go to the chassis, without a proper ground the line caps would basically be open circuit. On an AC/DC set like that black Admiral you tried the line capacitor is connected across both legs of the power line. On an AC/DC set you obviously can't connect a ground to the set so they used the next best thing by connecting a cap between hot and neutral, neutral is often still above ground but it's still closer then the hot side.
Regards
Arran