You see the same thing happen when theres a mass shooting and you get tons of people calling 911 and then you get a ton of people calling their loved ones to find out if theyre okay.
The cell towers also rely on an internet backbone so if you lose power long enough to lose internet then you lose cell towers even if they have all the power they need.
You could try putting all the communications infrastructure on a high priority circuit or something but unless your going to build out a hardened power grid parallel to the existing one, that would only cover you from load shedding not from natural disasters or even just power lines sagging on a hot day until they short circuit and blow the grid. So im not sure it would be worth the cost.