To your main point: I think you have it wrong. The first wave never really got solved, so you are seeing still this extension of a horrible first wave. It’s not even a question of a second wave.
For other countries, it’s different. I can tell you the reason in Hong Kong that they are having a second and now a third wave — government laziness and ineptitude. The government allowed 250,000 people to enter into Hong Kong and move around without testing, because they were part of some kind of exemption list that meant they needed to have free movement. These people included business people from China, ship workers from Philippines and Malaysia and Indonesia, as well as pilots and air crews who were from massively infected areas. You see, the way it works in Hong Kong is that there may be a scientifically sound reason why you would test these people, or put them in quarantine, or ban them from entering the city, but science doesn’t decide Hong Kong stuff. Groups of business people and “special interest” groups do. If they don’t want to inconvenience their profit margins, they won’t allow the government to do it. And since the government relies on them for stability, they just do what business interests tell them to. Except of course for things like hospitality and restaurants, which the government is now killing because they refuse to listen to their issues.