I think Q2503 is dead. Your measurement of 0.3V between base and collector is a strong indication. A silicon junction have about 0.6 - 0.7V forward voltage. Measuring a substantially lower value means a shorted device. There could be a short elsewhere. But this is unlikely.
Why would you say that? In circuit, it's entirely possible to have less than 0.7V between the base and collector since that junction is not forward biased. And these are darlingtons, so you're going to see a double drop measuring between base and emitter, so it's not always 0.7V.
RobII, I'm a little confused where you're probing the negative of your DMM in all these measurements, so my apologies in advance if there's a reading that disputes what I'm about to say.
Although it may be Q2503, I think it could also be the output of U2523, the latch that outputs the scan signal. I would check one more thing before you start pulling out transistors.
With your DMM set on *ACV* and negative lead to a ground (chassis is usually good), check the output of U2523 pin 15. That is the output that drives Q2503. You should see some reading there, indicating there's a changing waveform. You can compare this to the other scan outputs on U2523, pins 12, 9, 6, 16, 5. Those should definitely have an AC voltage on them since those buttons are working.
If you have no AC volts on pin 15, or very little (like < 0.1V), switch to DCV mode. If you get a constant voltage close to +5V or 0V, I would say that output on U2523 is dead.
If you do see a comparable AC voltage on pin 15, check to see if you see an AC voltage on the other side of R2536, which is also Q2503's base. You should also see an AC voltage on the emitter of Q2503.