I'm fascinated that there was a patent infringement warning (and an application form for licensing). Would a hobbyist building one of these things for their own personal enjoyment really have been in legal danger?

Hollywood itself moved far out to the west to escape Edison's patents, so possibly. That said, it's quite an impressive project to find in a magazine given the time.

I wonder what the purpose of that offer was? To locate technicians skilled enough to make or repair TVs, perhaps? To sell magazines? Or perhaps even to sell the parts they have on offer?

Making a Baird televisor with a Nipkow disk is simple, and a good hobbyist project. The pictures are dim, because only one pixel at at time is lit, and the disc is much bigger than the display area, so the screen will be tiny.

A build of a Scophony display, from 1930, would be a nice demo. Scophony, with its liquid "super-sonic light valve", was a much better scanning technology. Instead of one pixel at a time being lit, a whole line was lit at once. The company failed, and no Scophony displays still exist.