General question for everybody- if they got rid of the minimum wage- what is the lowest wage you personally would be willing to accept to do a job? $5 an hour? $5 a day? How low are you willing to go? If there was only one employer in town and you had to bid on your wage to get a job- how low would you go? If you do not go low enough, you do not get the job because somebody else is willing to do it for less. Lowest wage bid gets the job. Without it- you starve.
"Well, I would find a better paying job somewhere else" is not acceptable here. If somebody else has to accept $1 an hour, then so do you in this case.
Maybe a secondary but possibly related question- of those who would like to get rid of the minimum wage- how many of you currently are paid the minimum wage? It is easy to tell somebody they should be paid less- are you willing to be paid less too?
I'm a little confused by your question. You are making good hypothetical points to show why a minimum wage is a bad thing, but you seem to be concluding from these very points that it's actually good.
First of all, it is important to acknowledge that the scenario you described is completely imaginary and impossible in the real world. If there was one job available then all the population of the town outside of the one person who gets that job would then have to labor in various ways (albeit not "jobs") to survive. They would find that some of them are better at some things (such as gardening) and some at others (such as sewing). These relative differences in productivity are called "comparative advantage." They would inevitably arrange systems of commerce with one another where people focus on the things they do best to trade for the other things they need, which would be produced by the people that do those things best. They would use money as a medium of exchange. And, thus they would be employed doing work for money after all. The way to maximize the benefits of this is to keep the government out completely, having no regulations of any kind, allowing people to exchange freely among one another. If the government says, "No you can't work your garden and sell the produce unless you can do so at a rate that will pay you back at more than $7 per hour of your labor." then it would only make it harder on everyone, not easier.
But, for the sake of argument, let's pretend the scenario you described is not impossible. Suppose such a town actually existed. In that case, a minimum wage law would be very harmful. With a minimum wage law, that one employer would have to pay some worker minimum wage (he'd make sure it's the most productive worker he could find willing to work for that). But there would be other workers without jobs, some of whom would be willing to offer their labor for less. And even if they are less productive than that worker currently employed for minimum wage, they might offer their labor for a low enough price that they would actually produce more per dollar paid them for that employer than the minimum wage worker does. Also, since they're willing to work for less, that means they are more desperate for the job. So by lifting the minimum wage, the town would then increase it's total productivity and shift its employment more toward the person most desperate for employment. Since the one productive person in town is working for less, that means the thing he produces will now be available to the rest of the town (hopefully it's food) more cheaply, and they will be able to buy more of it.
You're probably right that most people here don't work for minimum wage. Therefore, we aren't affected by it in as direct a way as the people that do, or as the people who are unemployed now and would like to offer their labor for less than minimum wage if they were allowed to, or the people who own businesses that pay people minimum wage or would like to offer jobs at less than minimum wage if they were allowed to. Those people who are affected directly by minimum wage laws are the people who suffer the most harm from them. I am not personally harmed by them. But I'm close to someone who is, my developmentally disabled sister. She now works for a restaurant for minimum wage rolling silverware. Her hours are limited, her job security is low, and she gets no benefits. If profits of the restaurant go down too much, we know she'll be the first to go. She would love to work more hours and have more say over the hours she works, but she can't. If only she could offer her labor at a lower rate, then she could negotiate on other aspects of her job and ensure her security there. But she can't do that because self-righteous politicians have decided they know what's best for her better than she does (the dirty secret is that they don't really care about her, it's all about their union support). Several years ago she lost a job she loved at a supermarket because she just wasn't a fast enough worker to justify their paying her minimum wage. She ought to have had the option of offering to work for less, and nobody has the right to tell her she can't.