It doesn't rely on the power of government. It's a choice made by the workers. It gives workers a voice in labor relations. It is more likely to bring about living wage jobs than any action that the government might take.
I, too, thought the minimum wage was a good idea when I got my first job at 15 for $1 per hour (minimum at the time.). Finding an employer willing to pay $1.25 was even better. Getting a union job at $2.60 was better still. Unfortunately, unions have lost power in today's economy, and so have the workers.
That $2.60 in 1963 was the cost of ten gallons of gas. Four days take home would pay a modest rent. Where are the jobs that pay that much now? There are some, to be sure, but they're fewer and fewer all the time.