So, I got a 606 with sync and start/stop issues. It would always be in play mode, but the "RUN" LED wouldn't turn on. Neither the "RUN" button nor the DIN sync in/out would do anything.
Looking at the schematic, there's a flip-flop latch (IC2b) that seems to hold the play/paused status. When probed, no surprise, its output never changed. Working backward, the schmitt inverter that clocks it, never changes its output either. Back one more step and Q1, that buffers/inverts the run/stop input jack, doesn't change its output by much either.
Q1 is outputting a max of 1.3V, not enough to register a change on the schmitt. Time to try replacing it. The part reads "603 31F", conflicting with the schematic. I think it's actually a 2SC2603 as used in the Boss DR110, so I used one of those. The LED now works!
It's still stuck in play mode though. Looking at the schematic more, I realise the flip-flop doesn't directly tell the CPU if we're in play mode or not. It goes through Q4, SW1, Q5, IC3 and some passives. Q4 and SW1 seem ok because they're part of the path to the now working LED. So, Q5 was the first candidate. Turns out it had the same low-output problem. Slap in another new transistor and everything works!
Both of these transistors are directly connected to input jacks, so I suspect some external voltage killed 'em. These are probably prime candidates for replacement in any unit that has similar problems.