It can, when the government would learn to understand how to implement freedom, including its pertinent paradoxical rules, beforehand. The freedom to do whatever one wants, despite popular misinterpretation, actually does not lead to higher standards of freedom at all. No personal freedom can exist without its paradox to protect our collective freedom.
The government does not understand what constitutes freedom. Hence there is no way for it today to stimulate the strengthening of our health. An enhancement to stave off the apathy about our health, bad food, and other damning products aplenty, plus addictive medicine taking control of people’s lives yields an unsustainable rampage in costs for which we all pay a growing price.
Hard-hitting examples of our flawed implementation of freedom related to healthcare are mentioned here: What is the market size of health?
So, the answer is yes. The government can in principle provide a solution to the rising healthcare cost. But it would require a change in how we constitute freedom, away from our neanderthal implementation of freedom, towards freedom paired with our collective interests.