Is there ever any crypto project that allows user passwords to be encrypted into its private key? I've never heard of such I am just curious and thinking about a possibility.
What exactly is
user passwords?
If you are talking about password login details for websites, than this is possible with some hardware wallets but if you lose that wallet or it gets broken, you will lose access to your websites.
You can even use Fido 2FA with Trezor and ledger hardware wallets, but again if you lose wallet you lose access, so it's better to have backup device.
For password managers I would suggest using something like opensource KeePass software, instead of using hardware wallets.
If you want you can generate 12 seed words offline and use them as password for KeePass.