I am not sure how you can make the statement that higher minimum wages don't equate to job losses. Do you have proof? It's common sense that if you wage the price of something you buy less of it. Unfortunately for the poor, minimum wage jobs are rather elastic. In France, with a relatively high minimum wage, unemployment is double digits and places like McDonald's have gone to digital order machines as opposed to hiring more cashiers. While that might increase demand for computer machine manufacturing and services, the minimum wage students and unskilled persons aren't likely to have the skills needed to gain those jobs.
You also have the chronically unemployed in both the U.S. as well as places like France where it's livable to simply collect the government check.
Minimum wage shouldn't be a career. I have little sympathy for those who haves worked minimum wage for ten years or more. That suggests a lack of ambition, intelligence or both. Even the pimple faced 16 year old at Chick fil'a eventually gets promoted to team leader potentially in a matter of months or a year. Those who have worked minimum wage for 5 years or more, I'd be very interested in their work history, their criminal record, their tardiness, their productivity.
At a place like McDonalds, if you just show up on time consistently, you get raises. If you're a hard worker and express the desire there are plenty of assistant manager trainee tracks available.
Raising minimum wage isn't the answer -- taking away the incentive to be lazy would do more than throwing people an extra $2 per hour. The same bad decisions that got them in that situation are still the same bad decisions that they will keep making; unless there is incentive to stop.
Someone right now could go to North Dakota and get a job doing local truck deliveries paying up to $100k per year, some positions will even train. Working in the oil patch down in Texas and Louisiana offers high paying entry level positions that quickly end up paying over $100k.
My point is that there is always a way out of a minimum wage existence, the problem is that many people are content getting their free Medicaid, EBT cards and subsidized housing; so there is less incentive to move across the country or take a class somewhere or join a union apprentice program.
I am not saying end welfare or anything like that, but a safety net ought not become a way of life as it has for millions of people.