How do you improve the profitability of a lottery? Well, you don't really. Money goes in, some money comes out. You calculate the odds of winning, calculate how many entries there were, what the cost of the ticket is, etc. Once you have all that data, you check for winners. You pay the winner. You keep some of the funds for state-run projects. And you're done.
It's easy for a company to make a killing on the lottery. If you make $20 million on entries, you simply give the winner $5 million. He'll walk away ecstatically with his $5 million, and you'll just have made pure profit without lifting a finger. That's why it's state-run.
Now think about healthcare - how is it any different? It's an inverse lottery in that you don't really want to "win", because "winning" means getting sick. Money goes in in the form of healthcare payments. Money comes out in the form of paying for medical bills. Most of the time the payouts are small (think of it as getting 3 numbers instead of 6 in a lottery). Every once in a while, there's a "big loser" (as opposed to a "big winner" in a lottery). Someone gets badly sick, and large payments have to go to pay the poor soul's medical bills.
Like any other lottery, you can't really improve the profitability of the system. At least, not without cheating people out of their winnings (or medical payments).
So why is it that lotteries are state-run, but healthcare isn't? There should be no such thing as a healthcare provider, simple as that. The government should take care of it, just like any other lottery. An employer pays in a certain amount (just like they do currently to a healthcare "business") to "The Government Healthcare Fund". When someone gets sick and needs medical care, money comes out of the fund. Simple as that.
And what about the uninsured? Well I say we first try out just transferring the current ridiculous system to a government-controlled lottery. If it turns out that employers are paying 30% less for this new system (because the sleazy healthcare companies aren't walking away with a huge profit), then why not? If it turns out that the uninsured can be insured at less cost than what employers are paying now, who's losing?