Great work on the troubleshooting so far, it seems that you've covered most bases there! Just to clarify, when you mentioned reinstalling including deleting the local app data, did you follow the full advanced reinstall steps listed here? If not, I might suggest just trying the steps outlined there.
Something is rotten with 81.4.195 for new installations...
Today, I had a similar problem when I set up a new machine to dual-boot Windows 10 (1903) and Ubuntu 19.04. A fresh Dropbox (81.4.195) install on both OS's exhibited the same problem:
After downloading the dropbox-dist package and starting dropboxd, I was able to sucessfully connect the machine/OS to my Dropbox account. Afterwards the daemon status would display "Starting..." and then, after a moment, the daemon would restart. Then, after another moment, it would restart again. An so on.
In the case of Windows, the system tray icon would disappear and reappear. In the case of Ubuntu, I use the "headless" install method; the daemon would print "dropbox: locating interpreter", etc., pause, and then repeat. Both OS's repeated this behavior until I killed the daemon process. Neither OS made any progress downloading files.
Here's how I worked around the problem under Ubuntu by downgrading to 80.4.127:
# make sure that all Dropbox processes are stopped
ps auxwww | fgrep -i dropbox
# Remove 81.4.195
rm -rf ~/.dropbox-dist
# Download 80.4.127
# Install 80.4.127
tar zxf dropbox-lnx.x86_64-80.4.127.tar.gz
# start Dropbox again
# ... wait until everything is "Up to date" ...
# Upgrade Dropbox to the most recent version
At this point stopping and starting Dropbox appears to work correctly.
I found the 80.4.127 build here. I have not yet tried the same trick under Windows.
Thanks for your detailed post here @pvstodghill, I'm sure that this will be helpful for those who come across this thread! I'm glad to hear that your suggestion also fixed the issue for @simdhertefelt too, it's great to see that Community spirit in action.
Please do keep us updated with any other issues or questions about this, I'll be happy to help.