I don’t know what a PWA is, but as I wrote I just created a new HUGO site and installed a theme. Also, as I already wrote I did not have more than one server running.
However, the problem really seemed to be the browser cache. I was just surprised that no tutorials, official or otherwise, mentioned this at all. Most of them contained info about the auto-reload mechanism and the fact that the server updates resources on the fly, but nothing about the browser cache making things difficult. I have verified this on another machine, again with Chrome, and I get the same issues there. Luckily the extension makes it easy to disable the cache.
EDIT: Oh, and I could have sworn I tried with --noHTTPCache and it did not help, but now I am not so sure anymore…