From a lawyer perspective though, a cease & desist is a threat of escalation.

A lawsuit is the escalation. A cease and desist is a strongly worded letter that isn't really legally binding or anything.

The issue here is just impedance mismatch on the language. The legal department is used to doing things in legal terms, and probably sends out like a lot of cease and desists. And most of the targets of those also have lawyers who are speaking the same language, so it works. Just when it's a random individual getting the letter, there's a lot more confusion.