If a Cessna 172N has a factory original turn co-ordinator, it's electric. If it has a little red flag, it's electric. Those Turn co-ordinators are notoriously unreliable. They are DC electric, which means they have brushes which run on a commutator. The brushes wear, and the electricity stops flowing. Add to that, a commonly huge mess of greasy carbon inside. The red flag is not a 100% indicator of this failure. Listening is better, until you're in flight.
I have, and recommend to anyone who's going to keep the plane, to have the maintenance shop carefully remove it, and discard it with predjudice. Then, go and buy an AC version (which could run $1000) of the same instrument. You pay in the beginning, but it will very likely outlast the plane, with zero maintenance ever, and 100% reliability forever.
Or, if you're thinking auto pilot, now's the time, as some come with a replacement turn co-ordinator.
Though tempting, it would actually not be permitted to replace the electric turn co-ordinator with the older and more reliable vacuum turn and bank. This is because the of the design requirement for that aircraft that the turn indicator not be powered from the same power source as the attitude indicator - just in case one quits. Oh yeah, you know about that now! The vacuum pumps are a little more reliable that the factory original turn co-ordinators, but not much!
It's always a smart decision to consider the unknown causes of electrical problems, and not fly, if in doubt. My experience would have allowed me to continue the flight, as long as everything else was working properly. If one of those quits, my experience is 99.9% instrument (totally benign to the rest of the aircraft), 0.1% upstream problem (and that will be detectable by observing other things wrong/breaker popped etc.), probably still benign to the aircraft, but worthy of consideration.