Re: your nitpick.

It's true that TLS as implemented on the internet encrypts with a private key and decrypts with a public key, but this is not universal. Encrypted PGP communications, for example, work the opposite way.

Encrypting with a public key is a common technique if you want to ensure that only one person or device can decrypt it, which is what is desired here. It doesn't prove authenticity, unless you also sign the message with your own private key, but that's not the goal -- protection of the message from unwanted actors (and yourself even under compulsion) is the goal.