When using a password manager you don't need to ”check the input” or “correct an error” or “see what characters have been typed" (the only ”usability problems” mentioned in the article)

I think OP mean that if you use unique passwords, you would typically use a password manager and randomly generated passwords. You don't need to know so there is no point in seeing the password.

I don't think showing the password would actually encourage to re-use a password, but hiding the password may encourage to use a password manager and unique passwords.