Just to be clear, passkeys do not involve certificates or passwords at all.
When you set up a passkey between your client device and some website, your client device uses a random number generator to generate a very strong public-private key pair from scratch. Your new private key is kept secure on your device (requiring biometrics, etc. to get access to, as you noted), and your new matching public key is sent to the website and stored as a new authentication credential for your account.
So the security does not depend on the user coming up with a good password, and there's nothing stored on the website that could allow a web server security breach to expose a password or any other authentication/encryption secret. Public keys are meant to be public; they are not secrets; it doesn't matter if the whole world knows all of your public keys.
Now, if you allow your account on that website to continue to let you log in using just a password (as opposed to always requiring passkey authentication from now on), then your account on that website still has all the traditional risks of passwords. So, yes, a data breach of that website could still expose your password to the world. Or if a hacker got your password for that site another way, they could still log into your account using that password. So if you want to be more secure by using a passkey on a website, you need to disable normal (old, insecure) password-based authentication after you set up your passkey.
I do see from your comment on another Answer that you have questions around how the website would authenticate you before allowing you to establish a passkey. The answer is, if you have an existing account, the website authenticates you the way it always did (e.g. via password), and then after you establish your passkey, you can delete your password or otherwise disable password-based authentication for your account. If you are registering a brand new account, then you establish a passkey, instead of a password, when you initially register the account. There's nothing special about establishing a passkey instead of a password when you first create an account.