> buzzy harmonics (which weren't there before)
The buzz was there all along. It harmonizes with the hum, so your ear did not realy notice it until the hum was *gone*.
Yes, long grid leads suck up all the buzz in the room and amplify it. You need the cover tightly closed to verify buzz/humm measurements. If the wall-power supply is in the same box, it becomes a difficult problem. Shorter grid leads, shielded grid leads, are often necessary.
The other side: just like proper this-to-that grounding of input is needed, proper bonding around the power supply rectifier is needed. Rectifier should go RIGHT to first filter cap *before* any wires to the amplifier. I once saw (one of my ideas) with the rectifier circuit run all around the long side of the PCB to the first cap, and signal grounds tapped along it. It buzzed. X-actoing that trace and running a short fat wire overboard RIGHT to the cap helped a lot.