This is not true. Once cases are low enough a competent government can prevent outbreaks from spinning out of control, as has been seen in South Korea, China, and even Australia where new outbreaks have happened but haven't lead to exponential growth.
The problem in the US is that the governments in many states, and certainly at the federal level, aren't even trying to a competent response and instead threw up their hands. That's why they're seeing exponential increase.
A several month lockdown is necessary to get cases low enough that a competent government response and protective measures can keep them low even with increased economic activity.
You also have to clearly state what low cases mean. It's been almost four month and everything is random. Random opening. Random closing. But what I see are centers of power avoiding to declare something as their strategy, and share what they consider unacceptable.
The sad part is our incompetent government was supposed to get their act together in the meantime (more testing, etc), which hasn't happened.
Any country that decides to relax restrictions will see a surge in cases, not necessarily deaths, but cases for sure. That's true for sun belt States just as much as for Denmark.
Those advocating for low case counts at all costs are basically advocating to close parts of public life for good. Which can be a meaningful public policy position to have. But then say so. Don't talk about flattening the curve and hospital capacity. That's disingenuous.
If you are a governor or president anywhere, communicate clearly what your objective is. Are you working towards a kind of heard immunity situation while not overextending hospitals? Okay. In that case rising case counts can be expected. Some states are doing that. But say so. We are not stupid.
Are you trying to keep new cases below 10 per million and you hope to wait it out for a vaccine? Okay. Say so. In that case, I'll probably close my gym and skip that state for my concert tour. But I know what I'm in for. I know who is accountable for the decisions being made. You probably want to extend direct financial support to those businesses that you are preventing from opening. Otherwise they will sue you and eventually they will win, albeit might take some time for courts to get there.
The approach taken will be very different regionally. It's very likely that people in Texas have different preferences than people in New York or Michigan. They have a very different makeup and think and live differently.
What I'm missing is clear strategy communicated by governors to which their constituents can hold them accountable.