Socialized Medicine just means that we all get equal access to doctors... quality is something you have to work out for yourself, but these days, we have all sorts of ways of figuring that out.
In reality, that is not true. Just as people do not get access to the finest doctors in a timely manner in our Capitalist system, those depending on Socialized care get whatever the government doles out, the upper-income folks get their choice of doctors and clinics that accept only cash.
Just as with income, the same is true about Socialized healthcare.
In a single payer system, doctors cannot legally "extra bill" their patients or take payment in cash. Doctors have a fee schedule and extra billing is banned.
In regard to equality of services, the rich always do better under any system because they have more access to resources. Basic healthcare coverage provides you with a room in a ward. You can pay extra for a semi-private or a private room. A family friend hired 24 hour, round the clock nurses to manage their father's care at home when his Alzheimer's progressed to the point where he needed full time care.
The idea isn't to degrade all care to the lowest common denominator, but rather that everyone has equal access to quality care. If you take the time to know how the health care system functions, you can take greater control of your care. Lower income people may not have the education or resources to maximize their care.
For example: I'm currently on a 2 year waiting list for a knee replacement. It's getting difficult to walk, but I'm not using a cane or walker, and I don't have much pain, although it is starting to get annoying. Now, I can go online, find a doctor/hospital with a shorter waiting list, and travel to that city for the surgery. I'm also setting up an evaluation appointment with the Joint Clinic to find another local doctor to fast track my surgery. Money, computer skills and persistence will make it happen sooner rather than later for me. I would prefer to stick with my original surgeon because he's very very good, hence the two year wait, so I'm covering all my bases - looking for someone else to do it sooner, before I lose mobility and start having excessive pain, but you can see that in a system where everyone has equal access, the poor will not get the same quality of treatment that the rich can buy.
I have a friend who went for her first assessment last month and is having surgery this week, but she should have had the surgery 5 years ago and refused to have it at the time because of other medical issues. Now she can't walk, and she will not get a 100% recovery or freedom from pain.