Without a working filament to heat the cathode, the 6082 would not be conducting. The recommended resistance checks looked good. The builder is already out the cost and shipping of the new tube. He will ship the transformer back if you insist, but would rather you sent a replacement to spare the added expense. Our strategy with a new transformer would be to check the transformer alone, then do resistance checks of the rest of the power supply before connecting it to the transformer, then check HV DC before connecting it to the rest of the circuit. At that point, we would have a voltmeter connected to the B+ before we switched on, ready to switch off if it does not look right.
-Scott