The one part you forget in your description is that the purpose of capitalists is to pay the minimum amount of money while maximizing their personal gain. If they are allowed to (and way back they were) employers would pay mere pennies and make employees work 16 hour days. The employers did not really care if the employees could not afford the textiles or furniture build in their factories. Employers would always search for a way to pay even a lower wage such as hiring young children and women instead of men.
The minimum wage was put in place to protect the majority of the population against abusive employers. The minimum wage has not kept pace with inflation - from a pure monetary inflation perspective the time for $15 per hour is overdue. The other issue is that employers do not innovative when they can merely pay low wages. Increasing the minimum wage forces employers to make changes in equipment (driving manufacturing sales), increasing business productivity, dropping low-skill employees (creating the need for higher-paid skilled employees), and enhancing competitiveness instead of remaining stagnant and thirty years in the past.