There's a difference between noisy and biased. As long as the data is only noisy (that is, it has some random variations) I totally agree with you that it's fine to use, and that the results should be robust. However, if there is some sort of bias that only applies to a particular subset of the samples, all bets are off.

Just as a totally imaginary example, what if men with higher incomes are more likely to share their salary information than men with lower incomes, while at the same time the situation is reversed for women. So now you will end up with more reports of high income from men and more reports of low income from women, even if their pay distributions are exactly the same.

I am of course not saying that this is happening here, but these kinds of things would indeed completely invalidate the results.