I don't want to clear the birthday field from the contact. I just neither want to be notified of it nor see it on my calendar.
So... You don't actually care but you're annoyed that Google calendar makes you choose between admitting to not caring and getting 3 birthday notifications every day that you won't do anything about and don't want to be reminded about?
Somebody's birthday is interesting info to keep.
That doesn't mean you should get a notification when it comes up.