There’s a laser gadget available bigger shops sometimes have on hand to check for pulley alignment problems. A pulley could wobble left/right too I suppose, which this gadget would catch I think.
Ah! Glad you mentioned that, I’ll ask them if they have one.
It is normal for the tensioner to oscillate. Serpentine belts are usually long and stretch/retract with each pulse of the engine, this is very noticeable in the summer with the load of the A/.C compressor.
I have replaced hundreds of alternators and have not found the need to “shim” one, a replacement alternator should fit as well as the factory unit.
Your VW uses a double sided belt, they have used these for quite some time. You should replace the belt and see what you get. It may simply be that the old one is stretched out and the tensioner is just doing its job.
The new alternator, which is now charging your batt properly, will put drag on the belt and that drag will be shown on the slack side of the belt and at the tensioner, so it may be normal operating conditions with a functional alternator that is under normal load albeit with an older slightly stretched out belt.
Install a new belt and save the old one as a spare. Test and report.