For testing continuity between the CRT grid pin and the grid electrode a rather safe method is to test grid-cathode as a diode. You have to remove the socket to isolate the heater from the transformer winding. Supply 6.3 V to the heater and supply DC between grid and cathode with a suitable series resistor for current limiting (0.1 mA or so is enough). No other pins used. Continuity will show up quickly. IIRC 25 V is enough.
This way you don't have to do measurements in the presence of HV.
I hope for you that you can exclude a CRT problem!