Claude will happily accept secrets (passwords, tokens, ssh keys) as environmental variables or in files. It only refuses to handle them directly.
I have a local repo with my home lab mapped out and Claude manages servers via ssh. It is completely fine with using my ssh keys and pass.
Claude will also drive the browser. After a couple times, it will even offer to do something via the browser where no API or other machine interface exists.
And yes, the digital drudgery has definitely reached a point where this is worth the risk.