TWRP has lost most of its former usefulness on most devices because of Android moving on from several old ways without TWRP being able to catch up depending on the device. But if you want to experiment, you can find the vbmeta image the documentation mentions in the install ZIP files on the FP5 download pages … official or community (you need to know which variant you are running).
If the Fairphone 5 has video out of the USB C you can connect a “USB C Hub”, the one with HDMI and a few USB ports so you can connect it to a monitor and a USB mouse. Once you can get around in the phone save the data you need.
Someone did it at work the other day with just a USB C to display port (mini) but had no touch (mouse function) that’s why a full on hub is best. The actually come with storage slots too. I will post a picture shorty. I am on my laptop picture will come from phone.
Edit: The one I have has three USB A blue ports, Ethernet, HDMI, USB C power in TF and SD card slot. Pretty cool device, I can give my phone a wired connection if I want.
Thank you so much @AnotherElk@Jets@aibd for your very smart and promising suggestions! I explored all of them and eventually solved the issue after meeting with a peer who offered me to use her screen for a couple of minutes. Below are a few feedbacks.
Solution : with a working screen, make the FP5 seen by adb (adb devices) - it takes a couple of minutes - then, backup all datawith the non-working screen (if you can use the working screen only for a short period of time)
The key operations with the working screen plugged on my FP5 were the following:
USB Debugging: Enabled
Default USB settings: “File transfer”
Screen lock: from “Code” to “none” ==> important step: after getting the not working screen back, adb could not see my phone because of the pin code
Then, from my PC (Ubuntu 24.04) with my non-working screen:
adb pull /sdcard/ /my_backup_folder/ backed up all my data (photos, documents etc. + ./Android folder) but it does not save specific authentification data (e.g. you can backup Whatsapp Photos, Videos etc. but not the conversations if no backup had been made manually from the app recently)
App list: adb shell pm list packages -3 > /my_backup_folder/app_list.txt : useful for a future /e/OS reinstall
Contacts backup: adb shell content query --uri content://com.android.contacts/data > /my_backup_folder/contacts.txt : not a great solution (.txt file), but better than nothing if contacts had not been saved from the /e/OS Contacts app
SMS backup: adb shell content query --uri content://sms > /my_backup_folder/sms.txt : same comment than above with SMS
Calls journal backup: adb shell content query --uri content://call_log/calls > /my_backup_folder/calls.txt : same comment than above with calls journal