Pros and Cons list for using 3rd party login ONLY
Pro: user doesn't need to sign up, just logs in
Con: If your account is terminated by the third party for any reason then it can't be used to log in anymore. YouTube recently terminated peoples google accounts for using emoji's over and over again in a live stream after the streamer told them to do so for example.
Con: If the user closes their account on their own accord for that third party site they can't log into your site anymore either.
Con: If the third party terminates the developers account for any reason then no users can log in at all!
So yeah, if you're designing a website/app/anything with a login system, please please please always allow a user to create an account on your system too. It's fine if you want to give the user who don't care the option to log in via the third party, but I and many others will flat out not use your service.