It doesn't matter how you structure things you end up with some way to climb the hierarchy. The problem with pure communist systems is they try to prevent all hierarchies. Human nature attempts to circumvent this, the only way to do so is via corruption and political power. Corruption and political power exist in abundance in free markets as well, but there is another outlet for personal ambition. People can get rich. The hope is that you can build a system that incentivizes people to get rich by providing value for everyone. You can have private for-profit healthcare but you have to make sure that people get a good deal out of that. Normally we rely on competition to ensure customers always get a good deal. I don't think that works in this setting. Healthcare is tricky, I don't think we've found a good way to setup incentives for heath systems to have the best outcomes. We will need strong government regulation no matter what we do. Competition will not provide sufficiently good outcomes.
In the US you have great hospitals but at astronomical costs, the entire system is distorted by insurance companies and massive bureaucracies. Here in NZ the hospitals are terrible underfunded socialist messes. We can't hold on to staff because we don't pay enough. Emergency waits are hours. My friends wife just spent 3 days before getting emergency surgery to reattach nerves and tendons in her hand. She had to travel 3 hours to another city to get that care. The last two times I required emergency care the wait was over 8 hours. I can't pay for better service and there is no private emergency option.