Sorry, but a rich person going to a rich person's clinic 33 years ago didn't even prove a point 33 years ago, it's hardly going to prove a point now. People have to wait for care all the time. There are plenty of anecdotes about people having to wait for care in both socialized medical systems and capitalist health insurance scenarios. Turns out that rich and influential people can use their wealth and influence to get faster care. That's not some indictment of either private insurance or socialized medicine, it's just more evidence that money and power are money and power.
Back when the first President Bush threw up in the lap of the Japanese Prime Minister there was an attending doctor at his side immediately. That was 31 years ago and the healthcare debate was going on even back then. I remember a late night comedian commenting "That's the healthcare plan I want" about Bush immediately having a doctor at his side. It was just a joke because, obviously, no healthcare plan can give everyone a personal attending doctor to travel with them everywhere. When you use the rich and powerful travelling to get fast healthcare from the physician of their choice as your example for the problems with socialized medicine, it seems like maybe you have a problem with that.
The simple fact is that, if what you suggest is true, and socialized healthcare has catastrophic wait times that make conditions worse relative to the system in the US, then what is your explanation for why the outcomes are better under socialized healthcare? If socialized healthcare has so much worse wait times that make conditions worse, there must be some other factor that does an amazing job counteracting that glaring problem. So, I assume you must have some idea what that is?