You somehow forgot to mention that most (probably all) EU countries have laws that require you to know the birthdays of your customers - that of course overrides GDPR, or more precisely, the law is the reason to store the information so there's no need to find other reasons.

Also, don't forget that these laws also have requirements on you keeping logs, most of the time 3, 5 or more years. So yeah you have to obey a deletion request when that time is up, not "on request" - that would be illegal in most cases.

In many EU countries birthdate (and more) is public information, btw - my own birthdate is made public by the state itself (on the business registry website), together with my name and residence address. Same for any owner of real estate - be it land, house or unit - names, residence addresses and birthdates are publicly available in the online cadastre.