which is why I'm wary of using my password manager for OTP, and use a separate one. Not sure if it's too paranoid, but it doesn't make sense to me to keep the 2 in the same place.
Both seem true, and what to do to protect yourself more depends on what kinds of attacks you're interested in stopping and at what costs. Personally, PM + U2F seems the highest-security, fastest-UI, easiest-UX by far — https://cloud.google.com/security-key/