Something someone in private industry would do in an afternoon with a spreadsheet.
It's what? Maybe a dozen data points from each of all 57 states*, plus a calender date, and maybe half a dozen "other factors"
* 50 states, District of Columbia, Guam, Puerto Rico, US Virgin Islands, Samoa, Northern Marianas, Gitmo...
It’s a little bit more complicated than that. There are actually two labor surveys done each month - the corporate survey that the BLS does, and the household survey done by the private payroll processor ADP. So BLS is using data from two different surveys - the employee side (ADP’s) and the employer side (their’s). ADP surveys about 60,000 households, and BLS surveys around 140,000 companies and government agencies.
The ADP survey comes out a day or two earlier than the BLS survey, but the BLS uses the ADP numbers to determine the unemployment rate, since only the workers (employed or not) can say whether they are employed or not, and if not, whether they are looking for work or not (i.e, in the labor force or not).
Regardless, ADP does their survey during the week that has the 12th of the month in it, so the numbers are in. They just need to be crunched. Which, I most certainly agree with you, a private company could do quicker and likely more accurately.
BTW - I want to visit that Gitmo state. Supposed to be some nice beaches there...