I think I'm going to find it completely unrealistic how they portray society collapsing,
Probably, but keep in mind they have to make a show out of it.
I would imagine if society collapsed quickly and ZOMBIES all of a sudden became a real thing, the combination of religious nutjobs, actual nutjobs, bad people, protective people, and all sorts of other people would basically all destroy each other for survival.
If somehow communications kept up (which these days it seems people are willing to do ANYTHING to get on the air to report the news), this wouldn't be a hard problem to solve. When people die, headshot, cremate, done.
I don't think it would be all that hard to set up a system to contain it.
But, of course, initially people need to rally together and go around shooting all the Z's in the head. Again, I don't think that would be too hard to set up.
As long as people are still working at the electric and gas companies and the supermarkets manage to somehow get stocked for a few weeks, problem solved quickly.