I'm no fan of insurance/hospital bureaucracy, but at least I can go to any specialist doctor I want without having to see a GP first. And I pick the job and insurance provider I want to reduce the amount my family pays each year.
Emergency situations like the last one you mentioned, if they are out of network, are still covered at in network rates, thanks to my insurance plan specifically stating that in their plan documentation. I feel bad for your sister-in-law and am glad she survived, but she did have a not so great insurance plan. I would strongly consider getting a different job or lobbying hard with my workplace HR if I had a plan that would only pay 10% in true emergency situations.