I would take a look at the following links as I think they might give some insight into your issues.

https://support.google.com/chrome/thread/346896505/keyboard-not-working-on-macos-login-screen-when-using-google-remote-desktop

https://support.apple.com/guide/mac-help/control-access-to-input-monitoring-on-mac-mchl4cedafb6

I would also reccommend updating to the latest ndi tools. https://ndi.video/tools/