Webproxy WASN'T running. That was the netstat output from the WORKING nodes. Netstat had shown nothing for port 8080 on the non-working one. That should have been my first clue. Bu the lack of a generated log file clinched it.
Incidentally, I did look (and had tried) your suggested way of upgrading nodejs. For some reason, that method completely hosed the existing install on that server. As a result, I ended up having to remove and completely purge it.
Doing so also removed the DVSwitch installation (but, thankfully, left all the config files intact). After reinstalling DVSwitch, I ended up back at version 4 of node...which I was then able to successfully update the way I described.
I'm betting that it had something to do with the fact that the server is Ubuntu based (Mint) rather than "pure" Debian (Stretch). But hey, no big deal. Just happy it now works.
Thanks again for pointing me in the right direction(s).