Just to give an alternative point of view - I prefer third-party login whenever it's available, so the current flow works for me. I'm not saying you shouldn't add first-party login, just don't get rid of third-party login in the process.

I don't say that 3rd party logins are bad and should be removed, I just hate when website assumes that "everyone has Facebook, Google, GitHub, <insert some other trendy site here> etc." We fought long and hard to get rid of vendor locks as much as possible, why go back to these dark times?

My problem is that I can easily create account and store it in my Keepass, however when there are only 3rd party logins I have no idea which third party login I chose to use for a specific site. This sometimes means trying to authenticate with multiple third party vendors and sometimes accidentally creating a new account as I logged in with a wrong one.

Good points there and having done that myself as well I can definitely sympathise with you. Working on the email/password auth as we speak ;)