Chome extension never auto-fills (Safari works). Requires manual copy

50% of this was now self-solved! I thought I had gone through most of the suggestions on the website (including killing Chrome to restart), but when I browsed the https://chrome.google.com/webstore/detail/1password-extension-deskt/aomjjhallfgjeglblehebfpbcfeobpgk I found this suggestions by Nhat_Nguyen as a response to someone with my issues:

However, sometimes an update from 1Password or your browser could leave some pending updates. In that case, you can try these steps and most of the time, it should put things back on track for you.
Right-click on the 1Password mini icon in the menu bar at the top right corner of your computer > select “Quit 1Password Completely” > > Quit all browsers > Restart your computer > Try 1Password again

This was the trick to get the semi-automatic "Auto-fill" feature working! I had to kill the 1Password app for some reason. This makes it possible to click the 1Password icon in the icon toolbar and then press "Auto-fill".

Still, the extension does not auto-fill the entires on page load and it does not show any icons in the input elements. So this is only half-way solved.