If it were legal, wouldn't OSHA be involved?
How? Put a camera in every car, every building, every trailer, every alley and every bar in the country, 10 billion cameras watched by 10 million OSHA inspectors?
Still again, the topic isn't brothels. The topic is all of prostitution. Even in Nevada, very little prostitution is in brothels. Only weird old men go to brothels. Most prostitutes are too low grade and too much into drugs to work in a brothel, nor would very many be needed for how few customers there are.
Granted, you might not have any problem with everyone you know learning you hire prostitutes nor any problem if everyone knew you are/were a prostitution (hypothetical), but virtually no man nor woman want it permanent public knowledge they have gone to prostitutes or were one. So there also is the extortion potential no intelligent man or woman would risk.
It is only a very, very small group of fairly odd men who will go to a brothel, pay brothel prices or accept the rules and limitations of conduct at a brothel. Most want fast, cheap and dirty sex - over in minutes. In the car. Out of the car. In the backroom. Out of the backroom. In the alley, out of the alley. THAT is the overwhelming bulk of prostitution. For most men, sex with a prostitute will last about as long as jacking off to porn, if not shorter.
Explain how OSHA has any relevancy to that. It doesn't. Also, involving OSHA, even if it could, would just be adding massive civil and criminal laws against prostitutes. Actually that could be an SNL skit - an OSHA inspector getting into the backseat of a old car to a drunk man getting a $20 BJ, from a hooker flying on meth saying "excuse me, but I need to see you current license, your operation manual, confirm you are using OSHA approved safety equipment, your accident log and your activities log book."
There are brothels across the border in Mexico. "Boys town." I was down there once. Women in rooms with no door and open air windows anyone can watch from, Mexican federal police about with machine guns. There were 4 of us guys. Hell no. It was interesting seeing, the floor shows were fun to watch, the hookers coming up saying "f_ckie f_ckie?" Drinks were cheap. We didn't stay long.
Would YOU go to a government licensed brothel for sex? If you were going to go to a prostitute, would you go to a licensed brothel with security cameras taking your picture? Would you pay by credit card to make certain this is documented that way too? Really, would you?