The 3 companys I worked for the company men were God on the rig and could run off tool pushers or the whole crew.
I guess it depends on who you work for. An exception would be if some one called for Stop Work Authority for a Safety or Environmental issue. And that was just the last few years.
I guess it depended on who owned the rig? I worked on land rigs and the company man represented the company we drill for. As long as the company mans request was within the limits of the machinery and was safe, you did it. But, when the request was outside the limits of the machinery or not safe, you didn't do it, and there wasn't anything the company man could do. If you and the co. man disagreed, you wrote it up on the daily log.
The hole where I worked with 6 different co men was where they made bad decisions and I refused to do their bidding or suggested they do something different. By them refusing to do something different, it caused hole problems, not safety or exceeding the limits of the machinery, and they were run off, not by me, but the company they worked for. Never was my job or the crews in any doubt.
When I said I ran them off the rig, I dont mean I fired them, I meant I told them to leave the rig floor, tanks, etc. and dont come back until I said they could. They had a choice, leave the rig or I would stop operations. I only had to stop operations 3 times that I can remember, while I worked in the patch.
Never saw a co man run anyone of the drilling crew off.... I did see mud loggers, core drilling companies, cementing companies, welders, lay down crews, etc. get run off.