Advancing Cam Question

Why would the damper move at all?

You’re moving the camshaft in relation to the crank.

Edit-
Here's how I checked the accuracy of the 4* advance:
With it at TDC, I installed the chain in the 4* advanced position. and moved the crank (without moving the cam), until the gears and chain went on. I temp installed the timing cover and put the damper on far enough to be sure it was on the key. The timing mark moved 4* from TDC.

Got it.

It appears you’ve figured out that your test proved you installed the cam retarded.