First problem was that the battery was not charged completely and my MacBook Pro wasn't able to charge the phone's battery, hence the reboot loop.
I searched an original charger, connected the iPhone and let it charge. Once the battery was charged, the reboot loop stopped, but the Apple logo was still stuck on the screen.
Tried to force restart a couple of times, and enter in the DFU mode and eventually managed to activate the DFU mode.
Connected the iPhone to the MacBook. Initialized the update process, and while downloading the 6GB file, the MacBook disabled the USB accessories, therefore disconnecting the iPhone.
I connected to a fiber internet network, and repeated the steps above. The download was faster (6GB in 7 minutes) and the MacBook started the update installation process. It gave an unknown error at ~70%. I repeated the steps, it completed almost successfully, giving the same unknown error at 100%...
Tried again and got the same result.
Finally, I reset my iPhone and restored a recent backup.
Moral of the story: if the storage is full, you may want to delete applications and data you can get later anyways, rather than rebooting the device.