“Do city sewer systems continue to function if power is out for extended periods of time? What with sewage treatment and all, would the sewers get clogged and back up?”
Yes, some do require pumps.
This is one application where a septic tank and drain field is preferred. The vast majority of them (but not all) are 100% gravity operated.
With new regulations, most all out here are having move away from gravity operated to electrical pumped. No, makes no sense but it's what the goobernut demands. It has also demanded that water pumps be taken out of the river. If your well pump is an inch inside your water wall, then it's fine. An inch out, you're getting fined. Same water but somehow that's stealing river water. Never mind it's the streams in the hills that run down under the land to fill the river so it's all the same wet stuff. Never mind that river water adds nutrients to people's lawns which cuts down on dangerous chemical fertilizer run offs.