Sorry for beating around the bush for a while. I now have a better understanding of your situation. You are trying to use the unmanaged switch to “split” the Ethernet from the ONT into two? No, this won’t work because the Ethernet from the ONT is not NAT’ed. The Ethernet from the ONT can only be connected to one device at a time, so you should only connect it to the WAN port of the router in a residential setting. For other wired devices, you need to connect to the LAN ports of the router.
Given your description, the Other Room cable should be somehow connected to a LAN port of the G3100. Don’t you have multiple cables reaching each room? If so, try using one cable to connect between the ONT and your router WAN port. Then use another cable to connect a LAN port on the router to the switch located in your distribution panel. Then connect the Other Room cable to the switch. Then the setup should work.
In other words, leave the blue cable from the ONT connected to the Study cable intact. Then somehow route another cable from the Study to the same distribution box. Use this cable to connect a LAN port of the G3100 to your switch, which should be relocated to the distribution panel. Then connect other cables to other rooms to that switch.