And as a user, why would I trust my password to the website that rolled their own authentication over the big companies?
Other people have mentioned this, but if you're not reusing passwords, this shouldn't be a concern for you. Don't reuse passwords!
On the security front, companies that are implementing 3rd-party sign-in can still get hacked and leak your personal information. If that information is supplied by Apple instead of you, it's all the same. You don't automatically get better security because you're using 3rd-party sign-in, you only get better security if you're being forced to stop doing something bad (reusing passwords, enabling 2FA) or if Apple is providing less information than an account would ask for during signup.
To Apple's credit on that front, they do mask your email, which is a legitimate privacy improvement. But it would be better for that to be a generic service that allowed you to generate an anonymous email at any time for anything, rather than a perk that's hardwired into an anti-competitive scheme to make it harder for you to migrate devices or change services.
And as a user, what if Facebook/Google/Twitter/Apple decides you've violated their ToS, blocks you from your account, and now you can no longer log into any of the sites you've linked with one of those providers?
I know this is a bit extreme, and for many people, the risk is totally worth the reward, but I think that is one of the chief concerns many people have with trusting a third party for all their sign-ins. It's a single point of failure.
Do you use the same password everywhere, by any chance? :)
Thankfully, it's still fine for my bank login.
This situation is solvable by implementing forget password and storing user's email but many services don't and with a system similar to apple's where you mask the email or phone number, you can't do anything.
Even if the local auth system was poor, an email/password combo is simpler and faster without leaking data to social providers. There's no reason to remember which provider you used, or login to them first, or worry about loose permissions especially with future changes. It also limits the blast radius in case your social account ever gets compromised, and it's useful for completely anonymous and disposable accounts.