I can't tell if you are serious or not. Lets assume for a moment that there was once a benefit to BLS survey methodology ( I would argue otherwise, but w/e ). Is it a good methodology today?
So my main argument ( and frankly the only argument that should matter ) is that is a bad fit for the goal of estimating values ( even though we do know its failure modes ). Is that not enough?
You made the argument and provided zero supporting evidence. As it stands, it's merely an opinion, and appears to be an uninformed one until you prove otherwise. That's what people are asking you to do.
What are the alternatives, and do other countries labor statistics agencies use them?