First off, it's canonical that Pokemon, or at least some Pokemon, are aliens from space. Some are genetic/computer experiments, and there are definitely some early transhumanist themes in some of the games, but I don't think there's compelling evidence that anything is particularly post-apocalyptic
Second, there are farms, and food and drink are sold in restaurants. They aren't post-scarcity, but it is certainly true that we never see anyone really going hungry or wanting.
This is because the civilization in the Pokemon setting is a meritocracy where the primary governments are city-states ruled by the strongest Trainers, because, without Trainers, society would die.
Most, though admittedly not all, cities in Pokemon have a gym, which is where the strongest trainers live. They are the most important people in their towns at most times, and if a government official would be involved in solving a problem that occurs in the town, it is typically the Gym Leader who handles this. This is because the Gyms aren't just there to train new trainers; they're also there to serve as the town's governmental body and their first defense.
There are some well-maintained roads in the Pokemon universe, but you're right in that they're rare. This isn't because of an apocalypse. Rather, it's fundamentally impossible to maintain a road unless you can keep it protected from the regularly rampaging giant monsters in the area either through heavy application of Repels or constant defense. As such, travel between towns is rare for anyone other than Trainers, because they're the only ones who can actually survive the trip in most places.
This is also why 10 year olds go off on these journeys. People becoming Trainers is vital to the functioning of society. Without them, any town would be overrun or crushed the first time an Onix gets mad and rampages through. Trade between towns can't happen without someone to defend the transports from, say, a giant dragon attacking because you happened to go near it. Outside of that, Pokemon are regularly used in public works; they build and repair buildings, provide power, and even provide entertainment. Being a Trainer is insanely valuable, and in such a society, you're either valuable, or you're likely left for dead. It makes sense that families, especially those living in small villages, like the starting areas in the games usually are, would send off their children to become Trainers early. The child needs to prove their worth, and becoming a Trainer would mean a huge change in social position for everyone in the family. Most families probably consider it worth the risk.