What I need is:
show button on top bar if current website has stored passwords. At this point no logins/passwords are requested from the password manager, only website URLs (or even hashes).
After I click this button, request and show list of logins saved for this website. Let me choose login if there are multiple. Otherwise alter browser context menu.
Now when I right-click on text input, add items like "Insert login: vbezhenar" and "Insert password for vbezhenar".
And only when I choose "insert password" - actually request password from the password manager.
That's the only sane UI. Until I find it or write it myself, I'll continue to copy-paste passwords around.
It's a sad state of things where password managers are still not as usable as text file.