Hah, I just acquired a TDS-420A with the EXACT same issue. Everything else works perfectly, nice bright and clear CRT too. At first the Power/STBY botton did nothing, and the scope was always on. But after poking around, I now must hold the button in to keep the scope running.
I've been digging around in the scope for a few hours, using the service manual for a reference. According to the service manual, the STBY button signal is passed through from the button membrane "cable" to the FRONT PANEL PCB, where it passes through to the ribbon cable that connects the FRONT PANEL PCB to the CPU board. Unfortunately, TEK doesn't provide schematics and it's appears to be a 4 layer PCB making it pretty difficult to follow the power button signal.
Did you make any progress with your TDS-460A?