All of which would be easier with the correct manual, all i can find is the 608D.
I think the only differences is that the 608D has the crystal oscillator and headphone amplifier that is not in the C version, and the output stage is different. Perhaps the 608C does not have the buffer stage - you will work it out.
Edit to note: I'm not relying on the meters to tell me if i have an output, i put the output on my scope and got nothing.
Just be careful with the scope. You can destroy the scope input very easily especially if you are using x1/x10 switch-able probes.
Probes like this make your scope pretty safe:
https://www.aliexpress.com/item/100X-P4100-Oscilloscope-Probe-High-Voltage-2KV-Oscilloscope-Scope-Passive-Clip-Probe-Kit-100MHz-With-Marker/32850502604.htmlThe maximum voltage that ever reached the scope inputs is 2.25V with this oscillator - that is very safe.
To test the oscillator, you should be able to put the scope on the band switch contact going from the oscillator stage to the grid of the next stage. If there is a signal there, then you have a problem in the amplifier stages. No signal and the oscillator has a problem.
The oscillator valve gets its filament voltage from that DC mod circuit, and that could fail, particularly if they used a germanium PNP power transistor. The voltage on the transistor emitter should be about 9.3 to 9.7 volts DC. If it is 12-16V, then the transistor is shorted and the filament in the oscillator valve may have been damaged. Hopefully it is OK - that will be a really hard valve to replace:
http://www.r-type.org/exhib/aaa1026.htmThe valve came out at the same time as the oscillator, so the valve manufacturers may have designed it to HP's spec. Definitely not a standard valve.
Be really careful probing anywhere near the valve or the DC filament circuit. One slip could destroy the valve.