Thanks to the awesome troubleshooting help that I got privately through 1Password support (my first contact with them, very positive impression even though I'm just a private user, not a big business), I was able to determine that it's probably related to the fact that I have Lockdown mode activated on this machine and this Electron bug/behavior: https://github.com/electron/electron/issues/45837
As such it's not really an issue with 1Password in my eyes, and arguably intended behavior when running with Lockdown mode enabled (though intended by Chromium/Google). Therefore at least in my eyes the issue is resolved. If you're running into the same issue, then the options right now as far as I can see are: 1) Disable lockdown mode. or 2) Embrace your emoji-free life, knowing you're a teeny tiny little bit better secured against some potential attacks. #worth
Personally I'll be in the latter camp and whenever I need to reference the Emoji just open 1Password in Firefox. (I changed my profile picture here to one of the original SoftBank emoji to celebrate the occasion.)