I don't recall seeing economists that support a minimum wage do so saying it will create jobs and reduce unemployement. Even unionists admit that when they drive up wages they sometimes harm other workers. I have seen many arguments in support of it but not that one. You didn't really argue it, either, though, you just kinda stated it as fact.
It also drives up the cost of living and distorts the cost of goods. It's not simply aggregate demand that Keynesians fixate upon(especially Robert Reich, I think he has aggregate demand porn on his harddrive). Additionally, the thing that irritates me about this argument is that if it's good for the economy overall, why are we only raising it a little bit? How do you arrive at an arbitrary level at which you stop? What is a "living wage" for the entire country when the cost of living has such enormous variablility from state to state and indeed city to city? For instance, $250k may seem rich to Obama but New York's senators and representatives were none too happy when he wanted to raise taxes on those people because $250k isn't "rich" there. What do you do if/when prices start to rise again?
You answered your own question. We don't make policy based on "good for the economy". We make policy based on politics. Read above for arguments, political ones, arguing that some jobs shouldn't pay a living wage even if society demands those jobs be filled.
When I took Reich's class almost 3 years ago he was the first prof I had to show the paper that a minimum wage increase can increase employment, but he wasn't the last. I've argued it a bunch of times on this site and don't feel like repeating it all. I'd rather talk about raising wages in general.
More important is that you are asserting it causes inflation. But that's only if we push past full employment, which a function dependent on many variables but you can safely assume we're so far from full employment at this point in the USA/EU demand-side inflation is basically a myth. Notice the productivity-wage divergence graph above. That's a clue that massive wage growth wouldn't push aggregate demand beyond the limits of aggregate supply.
It will not work unless a company is compensated first. If there is no money to be had, then an outside entity is going to have to supply it, like the government, or the foreign interest who end up with that money because of investments.
The local demand for a product will not increase. People are not starving because they cannot afford the necessities, they are starving because a vast majority of their finances go to outside interest. Some of that money is not recoverable. If the government steps in and inflates the economy, all it is doing is narrowly inflating the local economy and the majority of the inflation still goes to outside interest.
The reason that hiring people and paying them more works is because labor does provide an esteem into a person life, that is lacking and being fulfilled by the waste of unrecoverable desires. The other problem is that corporations need to become leaner and not squander the opportunity that hard work provides them. Every one is entitled to a good time, but not at the expense of ruining the economy.
Well, just do it now. Or do it at the start of the next recession. At the beginning of recessions when budget deficits get super wide is when private sector savings go very high. You might remember all the cash US corporations were sitting on back in 2011 was coincidentally after a couple years of large deficits. We'd need another stimulus first to do it now, but we're so far from full employment we can afford another (big) stimulus.
Another way of putting it is, have an appropriately timed government stimulus first, like when we're below full employment, then mandate wage increases. It's a super winning, depression-busting combo.
I'm not sure what you mean by saying "a government inflates an economy". Do you mean inflates prices (inflation) or do you mean increases output? Usually "economy" means "output" which isn't something the word inflation is generally for.
Also I'm not sure what your point is about local businesses. More people buy flowers from the flower shop when times are good. Likewise, a small time plastics manufacturer from Missouri will get more orders from his clients around the country when his clients are selling more of their product that he helps construct.