Clipboard managers, such as the one that ships with Windows 10, allow applications such as password managers to mark something to not be stored in the clipboard history. KeePass supports this functionality as an example.
Yeah, my concern is that the user won't necessarily be copying from a proper password manager. Passwords are not the only information that people copy and paste: there are credit card numbers, SSN's, drivers license info, etc etc.
You could argue the security trade-off is worthwhile, and I might even agree if the clipboard manager isn't enabled by default. A clipboard manager requires you to treat your clipboard in a certain way that I don't think users do in practice.
It isn't enabled by default (currently) but there isn't really anything you can do to protect someone who copies important out of a text document and/or leaves open access to their desktop anyways without annoying them so much they work around it in an even worse way. Same thing with browser autofill and password managers.
Really though autotype is the safer option.