As for the CDC losing the trust of the public due to their flip flop on mask wearing, NYT article today shows that that is just not true:
"The poll shows that large majorities across the partisan divide trust medical scientists and the Centers for Disease Control and Prevention, though Democrats are more trusting than Republicans over all. The levels are similar to those found in public opinion surveys earlier in the pandemic, and in the years before it, suggesting that the politicization of the coronavirus response has not demolished the credibility of science.
In the Times survey, 84 percent of voters said they trusted medical scientists to provide reliable information about the virus, with 90 percent of Democrats and 75 percent of Republicans trusting the experts. Overall trust in the C.D.C. was 77 percent — 71 percent among Republicans and 83 percent among Democrats."
It does mention that some people have lost faith in the CDC due to their mixed messages, but overall, support still remains very high:
"In the face of those headwinds, the high levels of trust in scientists are remarkable. “I haven’t seen much evidence that trust in scientists or trust in the scientific community has eroded,” said Matt Motta, an assistant professor of political science at Oklahoma State University, who studies the intersection of politics and science."
One of the real problems is that people don't trust the new media:
"And, consistent with other surveys, trust in the news media is low, especially among Republicans. Only 7 percent of Republicans say they trust the national news media to provide accurate information about the virus. That could be a problem, since far more Americans are learning about the pandemic from news reports than from their personal doctor or the C.D.C. website."
On the other hand, "Pew research has shown that people who rely on the White House for information on the virus tend to think the disease and pandemic are less dangerous than people who get their news from the national or local news media."