Hello everyone ! I have my phone back after 4 days of decidedly unwanted dive into the recovery stuff. Now I'm sharing the result of this experience.

What is the recovery image ? If you are an experienced desktop Linux user, you most probably have already used a recovery image for it. That's a removable support based Linux distro that is running entirely in the RAM of the system to allow to examine and change the existing non removable storage of the computer: most usually test and install a new distro.

The UT image is the same, but as there is no easy way to connect a removable storage and boot from it in an Android type phone, the image is prebuilt and stored in a specialized partition of the phone, and it's started from the firmware - the menus that are accessible by starting the phone with the sound down button pressed.

As the recovery partition of the phone is small, the recovery image is limited. Also, it's Android based, as its name imply: the repo for it is
https://github.com/ubports/halium_bootable_recovery
I don't know where could be hosted a recovery for UT flavours without Android base (aka Halium), if there is one.

How to access the recovery image from a connected PC with USB ? Adb.
When the phone is in recovery mode, 'adb devices' displays the phone's ID and 'recovery'. Launch then 'adb shell'.

If stars align, you'll get a command line prompt on your PC, you are in.

Use 'whoami' to know who you are. If it's 'root', you are really in. Root is needed for accessing the UT partitions. If it's 'shell', you are a limited user. Exit and start 'adb root'. If it fails with a timeout, as is the case with my FP5 and my Kubuntu 24.04 workstation, it's not a worry: the root change has been done nonetheless, connect with 'adb shell' again and you should be root.

To access your user (phablet) partition, use 'mount /data'.
The data partition is directly linked in the fstab file of the recovery image. It's a classic partition, so it can be directly mounted. Note that looking at /data when it's mounted in recovery does not display the same thing as looking (with sudo) at /data when UT is working normally. You'll get something like:

android-data  cache  custom  system-data  systemd  user-data

The rootfs partition is stored into a thingy called 'system_a' and it's an Android thing. The real classic partition is called 'super' and it holds dynamic partition, while this is not a concept specific to Android (lvm is the equivalent for Linux), the implementation is Android specific. As far as I can tell, there is no command line tool that can be used to manage the dynamic partitions stored into 'super'. Note that Android stores other partitions into 'super', system_a and system_b most notably (system_b is used to manage smoother upgrades on Android), but the UT installer deletes everything except system_a (it uses another and very dangerous tool called 'fastboot'). Nonetheless, the 'super' partition and its lone dynamic partition 'system_a' is kept by UT and system_a can be accessed only from the advanced recovery menu, with an option called 'mount/unmount system'. Once it's done, the recovery image has access to the rootfs under /mnt/system.

In my limited case, the only use I have found of the system partition is to assert what is the rootfs version. Don't use /mnt/system/etc/lsb-release, it's not correct.
Rely instead on ls /usr/share/click/frameworks -la: the last one should give the current UT version.

The data partition can be used to get an emergency backup of the phabled user. Here are the incantations I have used to get a backup file on the removable SD storage of my phone and get it on my PC:

mkdir /mnt2
mount /dev/block/mmcblk1p1 /mnt2
cd /data/user-data
echo 'phablet/Maps.OSM/*' >fexc.txt
echo 'phablet/.local/share/flatpak/*' >>fexc.txt
echo 'phablet/conda/*' >>fexc.txt
tar -zcv -f /mnt2/sav.gz -X fexc.txt phablet/*

(back to PC)
adb  pull /mnt2/sav.gz
tar -xf sav.gz

The 'fexc.txt' is a way I have found to restrict the size of the backup as tar seems limited to 4 Gb files in the recovery image. There are other ways to do it but this worked for me.

Now to get the last boot log: same but it's under /data/system-data/var/log/journal. From the recovery prompt:

fp5:/ # ls /data/system-data/var/log/journal                                                                                                                                                                                               
28003ec6e17b4afcb8da051c6833e155

here '28003ec6e17b4afcb8da051c6833e155' is the randomly generated (by systemd) directory name of the journal files. If you look at it, it will of course be different on your phone.

Here I used:

adb pull data/system-data/var/log/journal/28003ec6e17b4afcb8da051c6833e155
journalctl -D 28003ec6e17b4afcb8da051c6833e155 -b 0

on my PC and the output gave me the necessary information to understand why my phone was staying put at 'FAIRPHONE' in spite of a successful flashing (in fact, several successful flashings of different UT versions...).