Even if I ignored people's human right to volunteer their own time for however much money they goddamn please, and instead imposed my own personal standards on what their time is worth...why would I support a national minimum wage? States and municipalities already set their own minimum wages which at least would theoretically better reflect the purchasing power and cost of living in their specific area. This is an issue that even lefties should agree makes more sense to settle on a local basis rather than from a top-down perspective.
Minimum wage, traditionally speaking, is not some arbitrary number pulled out of the sky. When done properly, it's a figure that is chosen based up how high it can be while having minimum negative impact on businesses. Doing this on a national level raises the purchasing power of the entire lower class, thus helping everybody in the entire nation in a short and long term.
Since we live in a market where laborers are setting the price of pay, not on what their competition is paying to keep their job attractive while being profitable but instead base what they pay on the shear minimum amount they can legally pay, many jobs are already paying much lower than they should from a true competitive market standpoint.
Example, Burger King and McDonald's are both profitable companies. In a truly competitive market, they'd have to raise their rate of pay in order to attract perspective employees, but not so high that hiring said employee costs more than their productivity creates in gross revenue. However, due to having multiple unemployed individuals out there, some actually very qualified to work at a job that is not fast food, they can pay as low as they want with zero disregard to profit margin considerations as people will still take said jobs because they need them.
Now, we know that a company is only going to hire the minimum amount of employees that it needs, because to do otherwise makes no business sense and causes a loss in profit. We also know that, due to the above labor market situation, that we have the minimum amount of employees hired at the bare market legal minimum rate hired. By raising the minimum wage a slight amount, first in consideration on inflation then after a flat amount, these companies will be minimally impacted.
Why? They cannot fire many people or cut back hours to save costs, as they've already done the numbers and are paying the minimum amount of hours necessary. Since they are already paying less than they can, as discussed earlier due to the labor market situation, raising the wage, while impacting profits will not damage the organization due to the fact they will still be, well, more than solvent. Prices of said goods shouldn't increase too much as, being in a free market, these locations are still competing with each other to offer a quality product at the maximum price allowed while still being low enough of a price to make the product a purchasable item to the consumer.
The reason why, federally, we'd want a minimum wage supported is the better my countrymen do in, say, Georgia and Texas then better I do here in Tennessee. The better my country does, as a whole, the better the individuals do within the nation.
Setting minimum wage locally just sets us up in a race for the bottom, which we're already in and raising it is already not going to have a huge impact.