Is there ever any crypto project that allows user passwords to be encrypted into its private key?
The topic title and this question are both a little confusing, friend.

Trying to give it a generous reading, and flipping your question around, it seems like you're asking:
"Can a private key be derived from a password?"If that's your question, then the answer is yes and the term you're looking for is "brain" wallet. One obvious technique (there are many) is to simply hash the password to derive the private key (e.g. SHA256("ChosenPassword")). Any sequence of steps that results in an integer > 0 and < N (the group order of secp256k1) will work.
Don't go rushing off and using this however, because the security of any such scheme is limited by the password, and people tend to pick rubbish (low entropy) passwords. Only risk funds on addresses generated this way if you really know what you're doing.