Just got recommended that video, LOL. What's with all the starting and stopping that poor lathe? You can clearly hear the interrupted cut and how it changes when you're cutting more than the weld. That's what I'd be focusing on.
First pulley is cast iron, and I have real questions about the stability of that "weld". I would have sat that flat on a graphite block and brazed it. I've turned a few pulleys, and getting them to run true on a lathe can be a royal pain. His mandrel approach was good, and even then you can see how the bore wasn't concentric with the grooves. He could probably have offset it (in a 4-jaw) so that the weld blended better by being on the "high" side of the cut, but while that may look better, it may also add vibration, so I do think he did about as good a job as could be done.
Second is aluminum, and is pretty much EOL and U'd out. That's why he was having trouble indicating it, but I think I'd have approached the angle finding in much the same way as him. In the case of it being worn out, changing the angle might even help with belt contact.