There is no public key encryption in Thunderbird 78. It was supplied by Enigmail until 78 and will be inbuilt in 78.2 expected in a couple of weeks. But there is nothing between.

If I want to send this guy an encrypted message I thought all I need is this public key, but Thunderbird always wants me to install a new private key. Why??

Do you have an own key at all? If not, then why sending encrypted messages to a recipient, when the response would always have to be in plain text?

Your own public key is needed, because the message also gets encrypted to that key. This is needed for you to be able to read the message after it has been sent. If you also sign the message, you do need your private key.