The starter sounds similar to the ones Honda uses. A radial misalignment between the starter and the cup seems likely to produce a screech. Since the air cowl on the vertical pull starter engines didn't need to be well-aligned with the center of the crankshaft, I guess anything could happen. If the outside of the cup is touching something, there should be a shiny witness mark that is easy to see.

Why a previous tenant would have converted a vertical pull engine to radial pull is a mystery. I have always thought the vertical pull starter was one of the best things about the third design series (i.e. a 2 as the second digit in the model number) vertical crankshaft Briggs engines - along with the automatic choke. Well, those things plus lasting forever.