The competition is largely not between mining companies, but between mining and other professions. In China, changing jobs pretty much requires government sanction. Thus you pretty well cannot quit being a miner unless some politician gives you an upcheck. Thus those running the mines have no reason whatsoever to maintain a safe workplace. You're expendable and replaceable and your choice is mining or starving. The US at least has the ability to go "you know what? This is way too dangerous. I'm going to do something else." Choice, first and foremost, helps safety. Imposing safety measures is often arbitrary, and doesn't pay attention to risk/rewards or whether a measure will actually make people less safe in the long run. People in a choice system make a decision which is better and worse. They'll abandon measures which are ineffective or detrimental, adopt ones that are beneficial.
The point of my statement is even this seemingly minor choice, love it or leave it, makes companies adjust to some extent. But! With actual free choice of companies involved, things could be even better. Companies would have to compete even in the same area with each other over how good their working conditions are, rather than just playing the high salary versus risk game with other industries available to the same workers. The Belters in EfT are already transiting to the different stakes. If a subset of workers are dissatisfied with any given company, there's plenty other rocks. And plenty of capital to go around apparently for establishing new companies. I wouldn't be surprised if the larger rocks had multiple grubstakes on'em.
If the belt were run like mining interests in the US, some government would take a look at the chaotic maze of different rock stakes, take all the mineral rights and assign them "efficiently", zone the rocks with residences, and people would be owing their lives to the company store quite often. The choice would be between a few companies, nobody would really be able to start them up, and productivity would actually be *down* because the company wouldn't mine more than it individually can handle the logistics for. They'd have less incentive to work on safety because they won't lose their force-driven stakes on the belt. To some extent they would still have to, because some might prefer to provide a service industry in the residential areas. Or move back out of the belts to terra or mars, since medical technology seems to prevent the deterioration of physiology that seems quite real in microgravity in EfT.
Compare to Chinese state-owned, and the gov would just lay claim to the entire belt, order mining with timetables utterly unrelated to any value for safety, and have no fear because you have to have permission to exit the dangerous mines. They may actually be penal punishments. Why would you care about the safety of said workers? Even should you be some sort of a weird, kind overlord, you would have to vary your measures arbitrarily, try to gauge slowly what statistically alters the injury and fatality rates. Innovation would be slow. Painful. And since they can't really leave, you may end up preferring to spend human lives to enrich your own pocketbooks. Or gain status in the government heirarchy. And ironically it's the least efficient of the three methods, since gov is the decider of what is and isn't viable for mining. And what should and shouldn't be mined first. And coordinating the supply chains. One organization, with all the limits on coordination of such a large operation.