All the things they beg to be free takes someone's time to produce.
It's only fair that if these people have to give their time for free, people wanting free stuff should start by refusing pay for their time.
But they won't because we all know it's all about "give me free" anyway.
when throwing more money at the doctors fails to motivate them
What do you mean by this?
Also you are free to seek treatment outside of a healthcare system. Having one in place doe not mean you have to use it, but to be honest I don't see why you would go outside of the system unless it was for some very specific reason like there not being enough specialists within the system itself.
What do you mean by this?
The number of people needing medical treatment can go up waaaay faster than the medical schools can churn out quality professionals. If the population-to-doctors ratio rises far enough, then they are incapable of providing the same high quality treatment to everybody. Rights are Equal, and an overwhelmed bureaucracy is anything but. Corners must be cut, thus violating the supposed human rights of the patients that weren't prioritized. Theoretically you can offer the doctors more money to do more work, but a doctor is not an automaton capable of delivering a consistent quality of care 24/7
I don't see how this is an issue restricted to socialised healthcare. A workforce shortage can happen regardless of how you handle healthcare.
I'd also add that if it is a violation of rights that patients are not offered equal treatment due to staff shortages, surely it is far worse to have a system designed around only catering to people who can afford healthcare. We have had issues in the UK with a lack of medical staff over the past few years, something that has oddly improved during covid because it raised interest in working for the NHS. It isn't because the system is bad, it is because it has been constantly underfunded for the past 10 years in an attempt by our conservative government to start privatising various branches of it.
It is, if and when throwing more money at the doctors fails to motivate them. At some point you work enough hours that getting paid overtime stops being enticing. So either you find an alternative way to motivate your medical professionals (do it, or else), or compromise on quality.
If we are to claim that healthcare is a human right, it impossible for that to be a right to receive treatment in a hospital from a professional; it would be a right to see to your own treatment. Self-medication, experimentation, medical tourism, "alternative medicine", homeopathy, quackery.