Ok, I think I figured it out. Yesterday before all this started, I was trying to figure out how to manage my config files (working to set up a tool changer and wanted a real system with git to manage the files). I was exploring how to essentially rsynch the files from my mac to the duet.
As part of this I enabled FTP.
I then started doing some test scripts to synch the files, including copying back and forth a bunch of tiles.
I then did a 'clean' install of 3.4.5, but of course it had an empty /sys directory and was complaining, so I copied back in my config. That config had a line enabling ftp. So, not quite the "clean install" I thought I had.
With ftp disabled, I don't see issues. I do sometimes see a single "connection lost retrying" cycle before the full page comes up, but it appears to be able to recover from it. Normally I don't see any retry, just a clean load.
Summary:
If you are having issues with Duet Web Control having difficulty connecting on a Duet 2 Ethernet (or likely Duet 2 Wifi), make sure you have ftp disabled. For some reason, the resources allocated to ftp (even if they aren't currently in use) make the web server much less resilient to any network noise.