This "UPS" solution is missing a key block though. A charging/BMS circuit for the battery. This would otherwise rely on having to manually attach a car battery

This "UPS" solution is missing a key block though. A charging/BMS circuit for the battery. This would otherwise rely on having to manually attach a car battery trickle charger (don't even get me started on 20after4's comment regarding battery composition choice...) every so often. But then using the original router 12v wall wart as the primary PSU would be unlikely to handle charging the battery as well. A cleaner solution would involve something more like: Mains supply->14-20v/5-10A rated PSU (such as a meanwell unit) -> BMS (battery management system) for chosen battery chemistry -> LiFePo4 battery pack (in a 3s config to give 14.4v nominal output, and however many times parallel to give the desired wattage) -> then feed into the circuit provided in the OP and then you have a fully self contained UPS for the devices covered. Though I might add a 5v/2A buck converter or two for being able to charge some USB devices as well...