Now back to the OP. In the picture there is a patch panel (basically a network socket with many connections). If I interpret it correctly, there are 5 individual sockets in the house. What do you get? Double or single sockets? I prefer double sockets, but with BT you're not always completely free in the choice. With double sockets, always make sure that 2 cables are fully terminated and not the good old sparing wiring, otherwise only max. 100 Mbit can pass through the line.
With a double socket, you could place the router in the living room. One cable connects the router to the fiber optic connection and the LAN returns on the second cable to the patch panel. There the switch distributes it to the other rooms. Now you can install a suitable access point for the router (AVM) on the upper floor. The AVM devices don't have POE, but are quite easy to set up for the average user.
With single sockets and the router in the utility room, you have different options in the living room. For example, you can place another FB in the living room. You then have 3 additional LAN ports for other devices. If you put an access point in the living room, you need a switch if multiple LAN ports are occupied.
As you can see, there is not just one solution, it primarily depends on what kind of wiring is installed and what your wishes are: only WLAN or a WLAN and LAN combo? Basically, it is important that the router or access point is placed as high and free as possible and not in or behind furniture, on the floor, or under cables.