After establishing with a voltmeter that the controls - timer, cycle thermostat, and high limit thermostat - seemed to be in good order, I replaced the gas valve solenoid coils. Only two screws to remove to get the front panel complete with door off the DG309 (don't even need to disconnect the door switch!) leaving easy access to the gas valve and burner. Some care required to seat the new o-rings properly and not to lose too much of the chassis wiring when stripping insulation for the new connections, but a pretty simple and straightforward installation. Dryer lit right up and the burner cycled properly again.
BUT as an amateur repairman I didn't recognize how badly the burner had eroded over the (37!) years and after three or four loads the burner nozzle simply fell off its mounting bracket. With the new part in hand it was plain to see how much of the original had rotted away, finally eating up the spot-welds holding the burner assembly together.
Again, a mere minute or two to get the front of the machine off. Unplug the igniter, one screw off to remove the assembled igniter & burner, one screw off and on to swap the igniter to the new burner, and the one screw back in to reinstall the assembly. Again accessible, quick and easy, but take care not to knock the flat igniter blade against anything in the process - it's fragile! Lit right up and now drying again like the workhorse it's been for decades