Hi Catweazle, welcome to the forum.
You might want to get the service manual and put the recorder in service mode (without cassette inserted) and use the “hole test” to check if the microcontroller sees that the switches in the cassette compartment are closing when you operate them by hand. You say you tested the switches for continuity with a multimeter; if they check okay and the service mode program is not “seeing” them, check if there is a problem with the wiring (check if the signal from the switch makes it all the way to the microcontroller). If the microcontroller is getting the signal from the switch (measured with a volt meter) but it’s saying it doesn’t, it unfortunately means your microcontroller is broken. But that’s VERY unlikely. Probably your problem is simply a switch that you overlooked when you did your continuity testing, or a loose cable.
EDIT: to get to HOLE TEST mode on the DCC600, power the machine on without a cassette, and push and hold REC/PAUSE and DOLBY while pushing PLAY. It should say “SERVMODE Vxx” (where xx is a number). Then push NEXT; now it should say SERVICE MODE. Then push RECORD to put it in HOLE TEST mode. Every switch you push in the cassette compartment should show up as “HOLE X OK” if I understand correctly.
It looks like the “CASSETTE IN” switch is part 1705 on page 63 of the service manual.