Message-ID is a requirement for Usenet where it came from.

It is a requirement for being able to reply to messages and in general for email threading.

Message-ID is a requirement to archive email.

Practically every email client has included Message-ID since dial-up internet was fast and fashionable.

Given all of the above I am amazed more places don't drop email without a Message-ID.

Not including a Message-ID seems to be saying you don't want replies and you don't want your message to be archived. That seems very shady to me.