In my low cost of living region you’d be hard-pressed to find any entry level positions (fast food, retail) below $15/hour. Panera (which needs staff at 4 or 5 AM to prep food for the day) are starting at $20/hour. Local restaurants and grocery stores can and do lose employees when a corporate chain raises their rates, so they have to keep up.
We could eliminate the federal minimum wage and very little would change.
And only idiots think that removing the minimum wage will make things better, but they are idiots, so there's no helping them.
If you eliminated the federal minimum wage then many states would immediately remove it without a replacement and let their people starve to death in the gutters fighting with each other to get any income so that they didn't starve to death in the gutters while raking in the votes from the middle class business owners who are raking in the dough because they don't have to pay their employees any more.
If anything, we need to make the federal minimum wage not be a set minimum but instead it should change and vary based on the economic health of the city, state, and county that the worker works in, and it should be enough that any person who works a full time job earns enough money to afford 1/2 of a 2 bedroom apartment's average rent + food, insurance, car, gas, and fun.
That would give incentive to local businesses to keep their prices affordable because lower prices would mean lower wage costs and therefore more potential profit.
You're right here -- we need a complex, multivariate analysis to determine fair wages across regions that accounts for average and personal economic health, transportation costs, food costs, housing costs, difficulty of the work, and even the prestige of the job and the mood of the worker.
This determination should be updated frequently, ideally in real time, so that the prevailing wages don't fall behind reality.
Even if this determination is governing the wages around you, you should always be allowed to negotiate something better for yourself.
When wages are finally determined correctly, then every worker should be able to say "this job benefits me more than any other opportunity available".
The US distributes plenty of food to poor people. My neighbors on SNAP get significantly more fresh food than they can eat each week -- they just throw out about half of it. In the presence of social programs, changing the minimum wage will neither cause nor prevent starvation.