Alternatively, he wants someone at the top who will create an organization that does not have to repeatedly restate massively incorrect numbers.

Such an organization cannot exist. These agencies are always balancing two opposing forces, timeliness and accuracy. Data collection is inherently delayed (e.g. a lot of it is from surveys that businesses complete at their own speed, or from reports that each state/agency submits on their own timeline.) So collection for a given quarter typically completes long after the quarter is over, and then it takes some time to crunch those numbers.

So if you want early data it will inherently be of limited accuracy because that involves a lot of extrapolation with whatever incomplete data has been collected by that time. If you want accurate data you will have to wait for it because that data takes longer to be collected. You do want both because you need to make timely decisions, since most times the early numbers don't get revised by much, but you also want to course-correct when later, better data gives a different signal.

Agencies like the BLS publish their methodologies in great detail. Big revisions have always been happening, only they are getting more attention these days because of the heavy politicization.

How massively incorrect are the numbers in comparison to previous years? Was it anything unusual?

Here, take a look for yourself: https://www.bls.gov/web/empsit/cesnaicsrev.htm#2024

If this is an understood part of the process, why is it such a problem now?

Name some organizations that have "fire employees until we get success". Does that create a culture that prizes success, or just encourage employees to hide failure?

The numbers get better as they get more data.

Or rather not mention them at all. He'd rather not bring attention to the topic to begin with.

I think you understand how labor statistics work about as well as Trump.