Trade in iPhone 14 Pro Max with third-party screen?

It's likely the trade in value for your iPhone has reduced to $0. When you trade in your iPhone, the technicians will run diagnostics to verify the parts are working properly and authentic.

Best case scenario, you MIGHT be able to pay out of pocket to have the screen replaced with an authentic part if the Store you go to offers it, then trade it in. But 1) Apple in most cases denies service for third-party tampered devices and 2) The screen replacement cost would be almost the trade in value.

I wouldn't use the mail in option and would recommend In-Store trade in where you will get an on the spot value that you can instantly use for the purchase of your new phone. That way you do not have to worry about an unexpected downgrade in value after you send it in. I know it is not possible for everyone to do the In-Store trade in depending on where they are located, but any slight inconvenience when it is possible, will be worth it.

I’d doubt you’ll get full value and may well get offered recycling, but you'd need to confirm any offer with Apple.