And of course, the external tool can have plenty of exploitable leaks unrelated to whether or not it’s integrated to some browser.
If the goal is to have better security, no method of using password alone will bring significant improvement to an authentication system, no matter how great the password manager it’s used with.
Which is why my password manager has zero integration directly with the browser, or anything else for that matter. There is a tiny little bit of extra legwork caused by this⁰, but IMO it is a good compromise between convenience and easily available attack surface.
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[0] and it might be susceptible to attacks that manage to listen to the OS message queue & clipboard where a browser integrated method would not be, but once something is that far into your system there isn't much that is going to help you except maybe an orbital nuke.