Hello, what you describe is the normal flow for Passbolt.
The password is only stored on your browser and could be different in every browser or computer you use to log in to your Passbolt account.
This is why you can’t log in to a new location. Passbolt extension needs the private key to decrypt your passwords and you have to provide it. It is also a security method to confirm that you are the only one who has access to your account.
You can read more about the Authentication that Passbolt uses here. You will note that is not a form-based login as usual and maybe this can help you to understand why you need your private key the first time: