Hi Dave,
I do remember an issue with the old TS-430 and the low pass filter relays but that would cause low or intermittent RX and usually a pop or two of the mic key brings the audio back. There was a fix for the relays but damned if I can remember it....hang on I'll have a scoot round the old internet....
Ah, found it! :-
https://www.qsl.net/wb4kdi/Kenwood/TS-430S/Service%20Bulletins/asb0902.html
If you've just picked up this radio it may have been languishing on someone's shelf for a bit so might be worth giving the old girl a workout. Cycle though all the controls, punching the buttons, twiddling the knobs and jump up and down the bands with a TX/RX cycle on a dummy load to rattle everything through and hopefully wake her up a bit. If you decide to 'clean' up the controls make sure to use proper switch cleaner and not the surf's-up wash with WD40!
I think one of the rear DIN connectors has a physical switching arrangement on it too (I'd have to look at the service manual) so might be worth plugging in and out an empty DIN plug to loosen that up.
There looks to be plenty of service tricks and tips online but be aware of the ones that suggest re-soldering every single joint or the usual 'recapping' solutions. You may well cause more problems following that route than anything.
Otherwise as you suggest it may be worth plugging in an external speaker to see if you can hear anything whilst twiddling the AF control and ensuring the SQL is off.
That should hopefully sort it but otherwise you might well have to put you 'rig-doctor' cap back on mate.
I wish you well with it there Dave and let us know how you get on.
All the best,
Victor