Any system will stop working when the battery dies. The point of saying POTS lasting through outages is because Telcos have to adhere (or should) to strict standards regarding availability of service and they maintain their centralized battery backup much better than a consumer does (or can).
I don't have any experience with VOIP, so I don't know how long their batteries last. However, given that people tend to use their smart phones for everything (GPS, video, audio, etc.) how much of a battery buffer is left at the end of the day to last a 24 hour outage (since power outages are generally unplanned). I know the smart phones I've used can handle a couple hours of GPS, video, audio, etc. and there usually isn't much battery left for voice or standby.
All I'm saying is that while centralization provides a single point of failure, it also provides a single point of maintenance and allows much larger battery backup than would otherwise be possible. Not to mention that it is much easier to restore power to every CO in the city to restore phone service than it is to restore power to the entire city (much like how blocks on the same grid as a fire station are usually the first to have power restored).