If you're working a minimum-wage job for more than a year, you're not trying hard enough. That is the whole of it. Minimum wage jobs aren't supposed to be lifetime jobs. They are a starting point to gain experience and a better-paying job. There's no logical reason to have a minimum wage. Employers know they can't get anyone for $2 per hour.
It should not matter where young people start financially, but it does matter that they start.
Many start as a summer job ( paying very little ), then return years later after they have finished school with a related degree.