Aman, check out the Health Care Reading List on Mises.org:
http://mises.org/daily/3737
I recommend you to read all the articles (there are over 20) and tell us afterwards if they could answer all of your questions.
You’re right about the fact that out-of-pocket payment would result in lower prices instead of insurance coverage. Some of the articles deal with the fact that government policy in WW2 led to third party payment… Good luck
