Closing schools and so forth was part of the flatten-the-curve effort. Really though, the CDC has been driven by the available science and by the reality of the pandemic, and when the CDC's guidance has changed it has invariably been in response to new data on COVID or to changing conditions (e.g. community spread, N95 mask availability, new variants).

It is overly generous to speak of "one side" as if there is another equally valid side to compare with. The CDC is not a political institution and they always cite published research in their guidance and public statements about COVID-19. The anti-mask/anti-vax crowd are going around spreading one fantasy after another, coming up with new and ever more outrageous conspiracy theories and excuses for refusing to cooperate; none of their claims have ever been based in reality, and that is why the story changes on a daily basis and why there are so many contradictory claims being made.

To put this in perspective, the high-level direction of the CDC's guidance has, since the very beginning of the pandemic, been to reduce the rate at which COVID spreads; the changes in their guidance have been about the details. The anti-mask/anti-vax crowd has gone from denying that there is a virus to saying that the virus is no worse than the flu to claiming that masks would suffocate them to whining about how mask mandates violate to their freedom and that they would rather die of COVID than comply with a mask mandate. They went from heaping praise on the last President simply because he was in office when the vaccines were first becoming available to complaining that the current President was being given too much credit for the vaccination effort to claiming that the vaccine is made from aborted fetuses, that the vaccine alters DNA, that Bill Gates inserted microchips into the vaccine, that the vaccine causes people to become magnetized, and that vaccine mandates are proof that the vaccine is the "mark of the beast."