|The writing is good and I was compelled to read up to the point where the limousine showed up. My first thought at that point was "this guy is finding every survivor and, personally, handing them a letter. How is he is finding them? Why would he care if this guy knew? Why wasn't he using a helicopter?" Then later "He found that guy and told him but didn't bother with the women? What a jerk." The idea of him finding survivors and then leaving them scattered, when he could be collecting them makes no sense. If you want to build a better world you don't start by having survivors starve to death. I hear him saying "Oh this will weed out the weak and toughen them up." HA. You don't get a better society by encouraging the very behavior you are complaining about: murdering, raping, and robbing.
So what is it you really want to write? A new Heinlein's "Farnham's Freehold"? A "Walking Dead" without the zombies? You just want a young dude to meet hot chicks in a survivalist environment? Those might be worthy goals but do you have some idea what your world looks like at the end of your story?
To eliminate a large portion of the world population of humans, bombs are not going to cut it (plus that is so 70s.) You would have to have the world's armed forces in on the game to fire off the bombs (which is, probably, on the order of a 3 or 4 million people that won't get killed.) What you really want is a series of plagues. They spread quickly, they don't require an army to start just a few dozen people in labs around the world. You want several so that fighting it is really hard (as one plague in Europe is stopped the completely different one from Africa arrives.) Of course something similar has been done, they have made at least two movies about it "Omegaman" and "I am Legion". I am sure behind them is a book (I am too lazy to look it up.)
The tough part is a plausible scenario where you have survivors yet get to your goal of "better" human being surviving. Small villages and towns would have to stop all strangers from visiting. At this point you have reduced the population but now have isolated them and got them to fear and hate strangers. How is that different?
Who are the instigators here? You've put in the "the governments of the world" are the ones behind it but that is difficult to believe for our world. Keeping something like this secret would be impossible. Remember governments don't build bombs, they pay outside, private contracting companies to do that. There is very little trust between governments so something where one is expecting the other to eliminate their own population, I don't see that happening.
If I am being completely cynical, the governments are the tool of the wealthy. The wealthy have no interest in fixing anything for the huddled masses. In fact they depend on them to continue to choose entertainment over education; to choose religion over science. This has made them wealthy and will continue to do so. The wealthy can choose who they breed with they don't care who Maude in Little Rock, AR gets busy with.
I hate to be so negative but if you are going to write a disaster-survivor story you need to try to break new ground. Interstellar broke new ground, even if, on reflection, it falls apart. They had a few good ideas (although I absolutely hate time-travel stories. I won't bore you with my criticisms of that movie.)
The game Fallout might be the direction to take. Instead of nuclear war have it be an environmental disaster where select groups of humans were put into underground facilities in cryogenic sleep only to wake 1000 years later than expected. I am just tossing out ideas, take or leave it.
Good luck and keep at it.