To clarify their issue, you can celebrate someones birthday without knowing the year they are born, you only need to know the reocurring date. You can't enter a birthday in apple contacts without a year, if you attempt it it sets it as the date of tomorrow.

My experience with apple is something's either a 2 minute fix or unfixable which to be fair is a reasonable way to do things though much less appealing to me (though less relevant for many users as stock android/windows continue to give users less and less control).