Is it even recommended to send them back to them for faulty device replacement though? I would not trust them handling my Ledger. Particularly with a closed source component they can mess with which may have easily stored information I definitely would not want it to.
You can't reset the device to factory settings with a completely broken display unfortunately and there is also no option to do that through Ledger Live. What you can do is to restore your Bitcoin wallet from seed on Electrum, for example, and then create a new wallet for temporary use until your hardware wallet is fixed. The old HW seed should never be used again since it was recovered in a hot wallet and has lost its protection.
This can become a lengthy process if you have multiple coins and you need to recover each and everyone of them and move them to a new wallet with a different seed.