Why would anyone go to a Burger King in the first place?

Also I don't normally nitpick grammar mistakes but you're using "of" when you should be using "have".

Fixed.

Right, but what if there are no financial incentives for Burger King under the free market, but Burger King's existence produces public "goods"?

That's where that whole argument against intervention falls apart. The free market needs judicious pushes in the right direction to optimize public benefit

Give me an example of something that can only function when run by the public sector. The only two examples that I can really think of are national defence and law enforcement. Even in these sectors, private sector can cover a wide array of services. The US military, for example, continuously turn to private contractors to handle catering, hospitality, etc.

I'm sure the food itself is worse then the flies...

Not necessarily outright run by the public sector, but funded by them. Let's take the argument for emerging industries for instance:

X-Corp in Canada is the only producer of regional jet planes of consequence. Y-Corp in Brazil seeks to enter the industry, but the infrastructure for this industry in Brazil is not there, and there is no free-market reason for the infrastructure to be there. If Y-Corp forces its way into the industry anyway, supply will rise, demand will go down, and prices will go down, meaning that the market has been harmed for both players by Y-Corp's entry in the short run, as prices still go down for regional jets altogether, and Y-Corp isn't getting much money.

Brazil chooses to subsidize Y-Corp, and now while prices have been forced down across the board, Y-Corp is making money while X-Corp has to eat the lowered prices due to higher supply. Since there is now a financial incentive for Y-Corp to continue working in this industry, it begins to build infrastructure, bringing a network of jobs in related fields to Brazil and expanding opportunities for other Brazilian companies (what we might call a semi-public good, purely business in concern but benefitting more than just Y-Corp). Ideally, of course, Y-Corp can eventually be weaned off subsidies and compete in an industry that is now viable on its own

A business that is out-and-out state-run could fill in for a business receiving subsidies in this scenario, but in either case, the market needs a push in the right direction. Obviously it cannot be undertaken recklessly, but the payoffs are there

Flies = extra protein

In order for Brazil to subsidize Y-Corp they have to tax somewhere else. While Y-Corps costs have gone down, the costs for other companies, elsewhere, will go up. This will mean that their investment will fall, as will the infrastructure development that goes with it. So, whilst Y-Corp is bringing in jobs related to its industry, another industry in Brazil will be suffering for it.

The difference between the two scenarios (where Y-Corp receives subsidies, and where it doesn't) is what determines where investment flows. In the case where Y-Corp receives subsidy, it is essentially a guy in an office making the decision, and his decision enforced by law; while in the free-market solution, Y-Corp success will be determined by thousands, if not millions, of micro-decisions - each taken because both participants will benefit.

You talk about how it will create thousand of jobs in the jet-manufacturing network, but fail to realize that this can only come by taking the jobs away from somewhere else. Ultimately, if Y-Corp want to make jets, they should relocate to Canada. If Y-Corp want to be successful in Brazil, they should focus on doing things that Brazil is good at.

Endless capitulation to comparative advantage on the national scale can lead to weakness in its own right, making veritable Cheetahs or Giant Pandas out of states, especially true if your comparative advantage lies somewhere downmarket, somewhere sensitive to shock like commodities. Why does Russia struggle to attract tech business to the Moscow area? They know the gravy train for oil will run out eventually, and is unreliable in the long term even though that is where investment should, under market rules, flow in the short.

A certain level of damage (via taxation) to businesses which do possess comparative advantage will not be truly harmful, so long as you don't void the advantage altogether, and with that extra cash in hand you can subsidize businesses that may prove viable and kick-start your economy into a higher sphere. This again is where we see non-market externalities that are good for the people altogether (namely advancement into modernity), which are beneficial, not of great detriment to the market in the short run (so long as your taxation is not crippling to your CA industries), and will benefit the market in the long run so long as you bet on the right horse. That last part, of course, is critical, and why these things should only be undertaken judiciously. Equally, you can't get sunk into what Latin America as a whole did (Import Substitution Industrialization), but the right kick can optimize the public good in a way that the free market utterly would have failed to do, benefiting a few Brazilian capitalists who just end up putting their money into Canadian regional jets with their comparative advantage, and no-one else.

There are blind spots that the market cannot account for, even in terms of actually making money and not in pure terms of public goods (like the environment or national security)

Law enforcement and fire departments?  Do you want a rent-a-cop system or one where selective payment of fire protection in a town happens, where some people in a town have fire coverage and others don't?  They had cases where they privatized prison systems, and judges got bribed.  There is cases where some things end up best in the public sector, as opposed to private sector, in case where if everyone doesn't get covered and pay in, things can be worse.  Disease control can be one also.

Anyhow, the original post had to deal with the belief than some people express where that the free market alone can end up causing businesses to fail if they don't do what is needed, and government doesn't need to get involved.  On that note, some who show up in other threads on here that get intot his, haven't posted in here.