I breadboarded the LED driver circuit using three green LEDs with -12V and -5V rails, and a 2N3906 transistor. With the collector disconnected, simulating a dead transistor, I get essentially what I got on the display board - about .5V at the emitter, 1.2V at the base, and a bit over 6V at the bottom of R11. And dim LEDs.
Then I connected the collector to the circuit, and the emitter voltage increased to 1.56V, the base became 2.26V, and the bottom of R11 dropped to 5.58V. And the LEDs became much, much brighter. In terms of current, I have 19.02mA through R1, less .12mA base current, leaving 18.9mA for the LEDs (up from about 6mA before, so no wonder it's brighter). It looks like R11 is passing 4mA, and the collector is providing 14.9mA.
It's pretty clear to me that the transistor has failed open. The EB diode still works, but I don't think there's any collector current. And by the way, whatever the function of R11 was supposed to be in terms of controlling current, one benefit it provides is leaving the user with at least some display, although dim, if the transistor should fail.
I think there's a chance the LEDs are still ok. I found a brand new 2SA1593T in my junque box, which is total overkill for what's needed. But I have no other use for it, and if I can physically remove the transistor from the board and solder wires to the pads, I should be able to wire this new one in. And I can worry about the LEDs after that if the display is still too dim.