I’ll repeat an unaddressed point/question I’ve raised more than once so far: do you have the service manual?
You’re having multiple signal flow issues with a console that has very poor user community support, because A. there aren’t a lot in use these days and B. they are not popular. I hope you get some responses from people who have actual past or present experience with them, but I think that’s probably unlikely, and outside of that the issues you are experiencing are all in a general category of issues that occur with old consoles. So you do the normal things which include, in your case, cleaning and inspecting the module edge connectors, cleaning/exercising pots and switches, and exercising any jacks with normalling contacts or making sure SEND/RCV jack jumpers are in place and exercise those…check for compromised solder joints on any jacks PCBs…you do all that stuff across the whole console and then take inventory of anything that’s not working, get out the service manual and start running test tone and probing/tracing out where you lose the tone and that’s usually where the fault is. Do you have a signal generator? If you don’t you can find a mobile app and use that since a low distortion oscillator is not needed for what you are doing. And then you need an audio frequency rated level meter…a true RMS measuring meter rated for relative accuracy across the audio spectrum. That or an oscilloscope. Do you have any of those things?
The Model 10 is not a complex device, relatively speaking. I know you are hoping your issues are specific and common the Model 10, but, like I said, I don’t think you’ll get that kind of feedback because your issues are just not uncommon things with any half-century old analog audio device, and more common the more budget-level it was. But definitely exercise switches and clean and inspect edge connectors…in my experience 90% of problems like you are having come down to problems with those two areas. And on any Teac/Tascam console that uses those jumpers at the SEND/RCV patch points, those must be present for signal to flow, they need exercised and, depending on the quality of the PCB base material, sometimes I find cracked solder joints where the jacks are soldered to the jack PCB…I don’t know for sure with the Model 10, but if the base material used for the boards in that console are the same as the 70 series tape machines contemporaneous to the console, then they are really thin phenolic resin type…the least desirable and reliable IMO and I’d definitely be checking for bad solder joints at jacks and edge connectors.
So start with those things and as I said earlier I’m happy to engage and offer some more specific support, but not without tech docs; schematics, parts lists, PCB layouts, and an overall block diagram.