Saying eye doctor in a conversation feels normal. Saying tooth doctor feels like listening to someone who had a bad disco elysium roll and they're punching themselves in the face to get the words out.
The one that always bothered me is Hals-Nasen-Ohren-Arzt. Why is it Nasen? We only have one. It should either be Hälse-Nasen-Ohren Arzt or Hals-Nase-Ohren-Arzt.