回答:
スクリプトを次のように変更しました。
(*
This script iterates over your contacts. For each contact, it will delete the contact that has the birthdays
*)
tell application "Contacts"
set peopleToChange to people whose (birth date) is not missing value
repeat with thePerson in peopleToChange
delete ((birth date) of thePerson)
end repeat
save
end tell