Yeah it was the biggest misstep so far and is what led to my mistrust.I agree with your opinion completely. However, them doing so and not admitting the real reasoning behind it leads to distrust of the CDC.
If they had just given the real reason, that could have led to vaccinated people putting more pressure on unvaccinated friends and relatives to get vaccinated because they could blame them for why masking was back. Instead, it led to people saying "the vaccines don't work."
They should have just come out and said “the unvaccinated are not following the rules so we all have to mask”.