Why did the birthday email not go out? Premium
This feature is part of the Premium plan.
The birthday, milestone and we-miss-you emails only go to students whose May get greeting and win-back emails switch is on (when editing the student, in the Automatic email section). These are greeting-type emails, which need the student's consent (for a minor, the parent's); the emails about passes and owed sessions do not.
Also check that the student has a birthday on record, an email address, and that No automatic email is not switched on. The Greeting emails filter in the students list shows who still has no consent.