I don't think a webpage can read clipboard without the user manually pasting.
If there's some malicious desktop program running on your machine at the same time as the password manager then you're probably screwed regardless of whether the password passes through the clipboard - but maybe there's some subtleties I'm missing here.
Between Keepass, 1passwword, and lastpass I've never seen a password manager that just shows the password without the user explicitly choosing to reveal it
