The reason they are doubtful is because other countries where they have done more to track cases between vaxxed and unvaxxed have shown that there is significant breakthrough. These countries are Britain, Iceland and Israel. Even states that have personally tracked this information have shown a greater than 1% breakthrough rate. Why would the US be such an outlier? As much as you don't like it and just want to trust, it's far more likely that the stat is just an asspull to get you to take that shot like they want you to. We are aware tha the US has taken great steps to obfuscate the reality of these numbers, they admitted it openly. That's why the data from other nations is so damning. They could have at least respected us enough to make up a believable lie. go with 10% for your failure rate, or 20%. It wouldn't look that bad and people would forgive it instead of being as transparently fake as 1%.
Also, they tend to have that average be calculated from back in January when they were still limiting vaccinations to just the oldest of olds as another way to cook the books. Yes, if no one under 80 was vaccinated, you would have a much larger pool of unvaccinated being cases. Even before the vax was available at all there were more cases in 30 year olds than 80 year olds, because there are more of them and they are more active in society.