Something I found very amusing: the average life expectancy was probably about 20, because 21's the absolute maximum and there would have been significant deaths from failing infrastructure. This means that under the UN's current criteria, the Human Development Index of this society is negative.
At the end of Logans Run, the evil computer is blown up and all the people set free! Everyone goes outdoors and sees the sky and the trees. This is an entire population that has been given free food, water, shelter, heat, and clothes for their entire lives. Now they're in the wilderness with no knowledge, tools, or even clothing heavier than lingerie. They're all going to die. This probably also happened every time Captain Kirk saved a population from a controlling computer.
Stealthily acknowledged: the credits imply their doom by running blended shots of the doomed people flying over the Carousel over shots of the burned-out remains of the city. At least the people Kirk rescued apparently still had their cities and machinery intact and a little Federation aid.
The books ran a similar scenario; while there was certainly a social collapse and reversion to barbarism (even more so), humanity muddled through.
On a more optimistic note, the old man must have some sort of survival skills and tools to have lived outside for that long though (quite healthily too, for someone looking like in their 60s). People can learn from him. Not a lot of hope but still quite some hope.
There is also the fact that they all can read, and books have survived whatever catastrophe wiped out the previous civilization, so they can learn from them too.
And of course there's the not-insigificant possibility that some other humans are still out there and can teach them how to survive. After all, that old man had to come from somewhere...
In the movie, Logan's destruction of a support column and overload of the Master Computer shouldn't have cause that kind of cascade failure through the whole city. However, remember that no one likely knew anything about maintaining the city aside from cosmetic appearance.