There is no market demand for unskilled labor at $7.25 an hour. Even if you removed minimum wage laws, you would still need to pay people enough to eat and stay functional in the workplace, which nowadays I imagine isn't much lower than $7.25, especially considering the only practical unskilled labor roles today are physical toiling, since almost everything else can be very easily automated away, so you need to fix their broken bones.
So I'd argue the open market in the US don't have enough demand for physical human labor at the bare minimum cost to keep people alive. The consequence is a lot of people are just not working. We have nothing for them to do at the given price that anyone is willing to pay.
So if the market has no unmet demands for toil, then guaranteeing a minimum wage job means you are wasting that persons time on some task that nobody needs done, at least not for the price he is paid, and you are not only biasing away the labor market you are wasting that persons time on some task that isn't valuable to be done.
Even if you have some market demand, the given rate is dropping annually due to the cheaper and more efficient automation of physical duties. I guarantee you once automated vehicles are entering the market and are legally allowed, there will be an absurd displacement of menial labor driving delivery trucks.
Other industries rife for that kind of displacement are farming (considering most farming tasks are procedural, automated farming machinery is practical, even if it needs high precision to harvest using computer vision), retail (with automated vehicles, you might as well buy all your goods online, shipped directly to your home, with no need for an intermediary store except in rare conditions like furniture and cars where you want to "sample" in person the goods), and construction (if you plant factory made homes, might as well build a foundation machine to excavate and lay a foundation without human intervention).
And then you have no use for human meat sacks moving their arms. We are already approaching that - it is why this problem even exists today. What happens when we get there?
Humans are not ready for the post-work era, and probably won't be for a few generations. When there truly is no more need for actual labor to make the world run, humans will have to reform their notion of "unemployed". As of now, we define people based what they give to the world, and so far the easiest way to measure that is through work. That's why it's such a huge deal that our unemployment rate is high, we have these immense conceptual blocks that we'll need to shed for the post work era.
In the mean time, we have to accommodate the current generations of people by subsidizing work. It's the only way to sustain our current paradigm and give people livelihoods, instead of leaving them desolate in the transition phase with callous explanations of technological progress.
You're describing a situation where most of the economic work of producing the necessities of human life can be done by robots (food/clothing/shelter/electricity/transportation of those goods to the humans that need them).
When that happens, the human race will be able to retire -- communism / Basic Income will actually work in that world, everyone will be able to get the necessities of life, and the small proportion of people who are willing to work for fun, or unnecessary luxuries that aren't necessary for survival, will be very small compared to today's pool of human labor, but still enough to maintain the robots and do the remaining jobs that robots can't do.
Offering money to drop out of high school would cause the drop out rate to increase further. Besides, I get the feeling that most high school drop outs aren't really looking for work. This is a hard problem to solve.
The money would be unconditional, not in exchange for dropping out of high school. You may as well say that soup kitchens are offering soup to drop out of high school and are causing the drop out rate to increase further.
If you are an economist (a field is already more an ideology / spin machine than science -- the science part in it is basically math), and you are more into entertainment (e.g making the NYT best selling list of shallow books) than science, then you can "prove" a lot of things, without actually proving anything.
Start with a few unexamined wrong assumptions, select the data that fit your conclusion, dress them up with anecdotes and prose, and there you go!