Your premise about what determines the level of a person’s wage is correct. The wage a person earns is based on his productivity and the value of the output he helps produce. As you say, in a market economy there is a spectrum of wages, those who have the greatest productivity in the most valuable goods earn the highest wages and those who have the smallest productivity in the least valuable goods earn the lowest wages.
What the minimum wage does, then, is criminalize the payment of wages earned by those workers with the smallest productivity in the least valuable goods. They can no longer be legally employed. This has no direct effect on wages and employment above the legal minimum. It simply criminalizes the employment of the least productive workers. The workers made legally unemployable by the minimum wage cannot compete with the employed workers because they lack the productivity to earn the higher wages commanded by the more productive workers. The jobs once done by the now unemployed workers will not exist, at least not in the manner in which they did before the minimum wage.
Of course, by forcing workers into unemployment, the output they were producing is lost and therefore, the standards of living of people decline. The burden of this decline is felt not just by the unemployed. To the extent that other workers were employed because of the demand for their goods by those now unemployed, they too might have their wages fall. They might even fall below the minimum and force them into unemployment as well.
This process of declining wages is only partially offset by rising wages in areas that now become targets of investment in capital. At least part of the investment previously made in the processes of production now abandoned because of the minimum wage will be invested in capital that will raise worker productivity sufficiently to sustain these new production processes. (A recent example is the robo-burger-flipping machine. A McDonald’s restaurant, perhaps, can have a work-force of three instead of six and serve the same number of customers.)