What you're talking about already exists, but few take advantage of it as yet. I finally opened a health savings account last year and it's the best health decision I've ever made. I would encourage everyone to look into them.
You don't need health insurance to pay for every doctor visit and prescription. It is for catastrophes only. An HSA is cheaper because it's paired with catastrophe insurance only, and whatever you pay towards your savings account is tax deductible. You should set up the insurance separate from the account, actually, so that if insurance rules change or your insurance rates change, you're not locked into a plan because they also have your HSA. If you contribute the maximum amount to your savings account per year, your tax deduction can be more than your insurance premiums. The account is yours forever, and you can do with it whatever you want -- keep it in savings, put it in a money market, or invest it in stocks (with ameritrade for my HSA). Before you're 65, there's a penalty if you withdraw it for anything non health care related, after 65, you can withdraw it for whatever reason, no penalty. So it is like an IRA, but without the income restrictions IRAs have. And every year they are making them a better deal. The deductibles can be higher (making the insurance cheaper) and the maximum contribution per year can be more. It's really the best deal going.
This is a true market reform. It is an option, take it or leave it, no government mandates that you must enroll. But it's such a good deal, most people who are at all financially savvy will take it.