Is there ever any crypto project that allows user passwords to be encrypted into its private key?
It's not crypto project, but few password manager let you encrypt the database (Which contain password) with private key. For example, KeePass have Key File feature[1] which can be used to encrypt and decrypt the database. Usually it's used together with master password, but you could choose only to use key file.
[1]
https://keepass.info/help/base/keys.html