Refurbished iPhone and warranty

Hello

I went to an Apple Store to get my battery replaced on my iPhone 13 mini. The warranty of it was already expired but I was happy to pay 80 USD to get the battery replaced. While replacing the battery, Apple technicians broke the screen of the phone. Not having a screen replacement they decided to give me a refurbished 13 mini. I was happy with that, I paid the 80USD and I left.

Then at some point I receive a notification that says me that the phone is out of warranty. I called Apple, they told me that a refurbished phone gets its warranty start date from the date that the replaced phone was bought.

I think this is a non sense. My old phone was just perfect, just needed a battery replacement. Now I've a phone which is refurbished (so I don't even know why was returned to Apple) and has no warranty.

What you people suggest me to do? Just let it slide or continue the discussion with Apple?

thanks

dan