My emphasis.
By the way, tempus,I misspoke about C71 being in the control grid biasing circuit of the 6L6s: it is feeding the 12AX7's plate voltages (thank you nigel for pointing that out).
But if you are correct about C71 AND C69 being tied together to a single cap, then the voltage at points "B" and "C" would be identical, shorting around the R122, R123 portion of the voltage divider network they make with R124 (not just a bleeder), and essentially putting the 12AX7s' plates (anodes) and the 6L6 screen grids at the same potential. Add that to a "leaky" (slowly, partially shorting as it heats up) C69 OR C71, or BOTH and you'll get the problem you describe.
AND, if the 47K you mention is R124 and it's missing (or replaced with a 470K) then the 12AX7 plates and 6L6 screens are going to be even further from design specs.
Finally, introducing the Fluke into the screen circuit of the 6L6s is, apparently, altering the screen grid bias, however subtlely, and allowing for correct operation.
Take the time to replace C69 & 70, as per the schematic (single caps for each is acceptable and considerably cheaper than a multiple can type), replace the 470K with the correct value and my bet is that you will have fixed it.
Lastly, paralleling a good electrolytic cap with a leaky electrolytic cap does not undo the effect of the leak.