There is a reason the S-Meter is misbehaving like this. Once the meter is zeroed, the needle should stay at zero when you turn the AVC off. It doesn't -- it pins negative. And if you zero the meter in the AVC "Off" position and then flip over to Slow, Med, or Fast the needle swings positive with the RF gain at zero.
You can see it happening in the numbers. The meter amplifier tube is half of a 12AU7. Here's the data:
Plate (Pin 1)- Off: 81v; Slow: 84v; Med: 83v; Fast: 83v.
Grid (Pin 2)- Off: 0v; Slow: -1v; Med: -.8v; Fast: -.6v
Cathode (Pin 3)- Off: 3.3v; Slow: 2.9v; Med: 3.0v; Fast: 3.1v
This is exactly the behavior I would expect. Look at the simplified sketch attached. The circuit is a pair of voltage dividers. The variable resistor is the tube. The sensitivity is shown as fixed. The zeroing pot is shown as two resistors for clarity.
The meter reads zero when the two voltage divider center points (red dots) are the same. The zeroing pot trims the one to match the other.
The lower part of the fixed/right hand divider has four settings by the switch. This is to compensate for the four different values that occur on the upper part of the left hand divider as the tube's DC operating point changes due to the bias changing with the switch on the grid circuit. The two sets of changes are supposed to track, so that the potential between the cathode and the wiper of the zeroing pot stays zero. What's happening is that the two changes are not tracking. Something is off with the bias part, the divider switching part, or the element the bias part controls (tube).
HOWEVER:
it may never have been perfect. And, if in your troubleshooting you've changed the meter sensitivity to a significantly higher than original point, the problem may be magnified.
I would also note that old resistors with 5% tolerances were generally much closer than 5% when new. When I poke around in 50's to 60's low power circuits (i.e. no heating from current flow) I am usually astounded by how right-on the resistors are. If yours are only "within spec" it may (or may not) be good enough...
Ed