If you store the key in Bitwarden or Keepass, what makes it different from a password?
Also since the service does not store your private key, it is more resistant to data-breaches as that is one less potential breach source.
Heck, the company could easily say that your password needs to be 128 characters long and use multiple types of characters - and tell you to use a password manager (that both generates and fills in that information for yet).