FM MUTE will only work effectively with an antenna connected, and with you able to get both weak and strong signals. FM MUTE is a cut off for weak signals and inter-station hiss, not a mute switch that you would associate with dropping the volume so you could say answer the phone. It will mute any signal that is below a set threshold when on and also inter-station hiss. It is not adjustable.
I agree with rcs to check voltages on the Power supply against the scat 1st and address any problems there. Think of the power supply like your car engine. If it's NOT running on all 4-6-or 8, it's gonna run like shite. When doing one of these it's best to do the power supply 1st, then the Main amp, then start at the inputs and work towards the amp. Replace any problematic transistors (2sa726,& 725's and their complimentary partner if any). Trouble shoot the protection board, (look at sx-1010 rehab projects also as the 939 and 1010 share the same protection board. Everything there that applies to the 1010 applies to the 939.)
When checking the offset and bias on the 939 while on a DBT, when finished, turn them DOWN all the way. You'll fry the amp if you don't due to the increased voltage. then readjust when on full line voltage. Set it for 25ma as per the 1010 update. 50ma is a mistake the PIONEER Engineers made in their math and Mark The Fixer found it and explained why it was high. I would get new pots for the bias and offset. Like RCS said the old ones could be and more than likely ARE flaky.
Check the old pot with 2 dmm's set up on the correct pins. Rotate slowly and check operation one side against the other. Shoot some DeOxit F-5 (F-5 is the Faderlube not F-4) into the pot and work it back and forth 50 times and then check again. If it's now running right keep it for a spare (sealed in a plastic bag). Do the same for the new one. Note that some PIONEER Volume pots do get offset. See if you can get inside (take it apart) and check the wipers for play on the shaft. You might be able to work out the play, or not. The only place to get factory pots is a parts unit. So you're kind of stuck trying to get and keep them
working. A lot of manufacturers used the same brand parts as pioneer and have the same problems. So it's not an isolated problem. A lot of people just chalk it up to the age of the unit.