Here's the secret though: It's their ass on the line if it doesn't get done. If they're a middle manager, they now have to go back with hat in hand and tell their higher up they made a mistake. That repeats for however many layers of middle management exist until resources get properly allocated. It's their business that takes the hit if it gets out to client's that project their having built is meat grinding their laborers.

You the end laborer, hold the power in this case. A manager cannot knowingly force you to conduct your work in such a way as your rights to a safe workplace are jeopardized. You have the power to expose it, and make it unequivocably known to your higher ups that these violations are happening. You can report your employer to your local OSHA enforcement arm.

A law on paper but never enforced is no law at all, and a workplace where everyone lets the boss abuse one another is the only outcome if you won't stand up for you, and sometimes, encourage someone else to stand up for themselves. I did it for my people damn it. You need to do it for yourselves and your peers. This is why labor organization is important. Businesses can abuse you because people don't act in concert to the degree the business does.

No work is so important it cannot be done as safely as possible. No one has the right to conspire to put you in harms way. An employer who provably does is not entitled to be an employer in the United States. That is a privilege, not a right.

Fight, damnit! Fight! Or vote with your feet by finding something else to do! If you don't set the bar for people to not work for, and refuse to make their life more difficult by making them accountable for their own malfeasance, you're passing the buck to the next crazy motherf'er who will, and we are in short supply.