iPhone 13 Pro coverage expired, no service history

Well, the iPhone 13 Pro was discontinued by Apple in September 2022, so if your phone really is "new", it sat in a warehouse somewhere for 3+ years, probably longer since it is unlikely that you got one of the last phones off the production line in 2022.

This would have been terrible for the battery.

Since the warranty has already expired on your phone, this tells you that you did not buy a "new" phone. You bought a "used" phone, or it might have been "refurbished". If the seller told you that you were buying a new phone, you need to return the phone for a refund.

If you decide to keep the phone, you will have to pay for any service that is required on your "new" phone.

Check Settings > Battery on your phone and the tap on Battery Health & Charging. If the maximum capacity is 80% or lower, the phone needs a new battery. About $90-100 (U.S.).

From where did you purchase the phone? It was discontinued more than three years ago and is decidedly not new.

Um.. no, you didn't. You purchased a gray market / used / possibly refurbished device that no longer has a warranty.

That model was discontinued in September of 2022.