I have some news for future reader…
I solved my problem!
In this thread Bluetooth problem on Intel Dual Band Wireless AC 8260 for a dual boot / Kernel & Hardware / Arch Linux Forums someone pointed out here drivers - Bluetooth arbitrarily stopped working on Ubuntu after dual-booting to Windows - Ask Ubuntu
I didn’t understand why Windows was related…
In my case I don’t even dual boot with Windows…
In fact I even never installed Windows on this machine…
What I did I simply disabled the BT module on my BIOS…
Then booted… And restarted a couple of times (with obviously no Bluetooth)…
I also noticed that the computer booted somehow very fast (more on this below…)
Then I re-enabled the BT on my BIOS…
Then booted again… (which did happen quite fast also - see below).
At this moment the BT worked…
However, before I tried to disable/enable the BT module on the BIOS I did the following.
I changed the content of /etc/modprobe.d/iwlwifi.conf from:
options iwlwifi bt_coex_active=N
to:
options iwlwifi bt_coex_active=1
But this changed nothing… Even after some reboots and re-following the steps on the post above - loading the module with modprobe etc.
So I’m not sure if this changed actually contributed to solving the issue alongside the BIOS enabling/disabling.
Edit: I tried again by resetting this option to bt_coex_active=N and the BT still worked but my boot time (but broke my boot time again). So this was not related to the BT issue itself.
PS: Also about my booting time (which I was talking on this thread Big delay (1/2min) upon boot before system responds )…
It seems that it also solved that!
Now the boot time is not really that fast compared to what I saw on some computer (and even on my previous laptop - which I don’t really understand since this machine is very most powerful (TR1950X, 128Gb with nvme SSD).
But before this “fix” the computer was taking more than a minute so I’m still happy…