Anonclassics

From a practical standpoint, you are not going to have time to pick up the basics of Greek or Latin when you start a PHD program. That's why programs require students have taken 2-3 years, if not more for philologists. You need to be able to hit the ground running and start tackling that reading list, as a fair number of archaeology programs will require you to pass a reading exam in one of the ancient languages as well as in the modern languages. You will also have a zillion other things to do because you're studying archaeology. For your own sake and health, you do not want to be behind in the languages. By the way, you want to go to a program that requires archaeologists to have some training in the languages. Good luck getting a job in a Classics department without that.

Being a better professor* is putting the cart way before the horse. The ad comms want to know that you can handle the ancient languages because you need to hit the ground running with grad-level classes and preparing for exams. It's purely practical. *I'm going to stay this even though I know it will not be well received: the chances of becoming a professor in Classics or one of the related areas is incredibly small. Please think about the very real possibility of spending the next 5-7 years of your life pursuing Classics and then not finding gainful or viable employment.