It may be true in your country that the top tax payers only pay double of what the bottom tax payers pay, but that's not the case in the United States.
http://www.ntu.org/tax-basics/who-pa...ome-taxes.html
http://taxfoundation.org/article/sum...ome-tax-data-0
The top 10% of earners pay 70.5% of the total income tax for the United States and the top 50% pay 97.7% of total income. Even if somebody making 500k a year paid only 25% income tax that would be 125k. Somebody making even 125k a year will not be paying half of their income in taxes. So the wealthy definitely put forth much more than twice the amount of others.
I agree that socialized systems and private systems aren't mutually exclusive, but this debate to me is about how much money you put into socialized medicine.