This poses the problem of having to determine and impose the standard of what is proper compensation for work. Moreover, when you hike the minimum wage, you're changing the relative price of some factor involved in the production of goods and services without making anyone more productive.
Let me just try to get what you have in mind, so talking becomes possible. You seem to assume the lower pay less skilled workers receive is the product of a negotiation process that favors employers. You suggest using price control to tip the scale back in favor of (some) employees. Is that roughly what you have in mind?
While it is unclear to me to what extent we can in the abstract determine exactly what is the labor of someone worth, I would assume an employer wouldn't hire you or would eventually learn not to hire people who cannot help them turn a profit, so there's at least an upper bound on what they will pay you, regardless of whether or not there is a minimum wage: it's what you can add to their bottom line, essentially. Now, suppose we do step in and considerably hike the minimum wage or some 16, 17 or even 18 dollars an hour. As you noted, it's possible some investment projects will become unprofitable. You brush it aside saying that maybe these projects shouldn't be pursued in the first place. However, you have to note that it doesn't expand the space of profitable projects to legislate a new price for low skill labor. It shrinks that space: some projects which would have been put into effect will not materialize and some existing projects might also become unprofitable and be terminated. This can take place wholesale, but also incrementally: businesses may cut opening hours, reduce full-time or part-time positions, or turn some full-time positions into part-time positions. The other aspect of this is the relative price change. Now, to the extent that it is possible to do it, employers might try to substitute low skill labor with higher skill labor or machines.
This might seem like an oddity but think about teenagers looking for a job. If I want to hire someone who never had a job, there needs to be something to help me cope with the costs of providing training. It is also the case that the teenager might cost me some business by not interacting properly with clients, or by making mistakes that force me to pay for replacing certain things. Other types of mistakes might also force me to pay for more frequent maintenance of the equipment we use. It's not a college educated lawyer that you will hinder by enforcing a 15$ minimum wage and, although it might serve some people, it also prices some teenagers out of a good learning experience.
None of these scenarios are mere fantasies and people can probably react in many more ways I have not anticipated above. Of course, you may still object that a small paycheck is just too small, even if you sometimes would implicitly end up doing it on behalf of people who would take it anyway. You might also think that something is missing in the above story and that, in fact, more people will find working hours after a larger minimum wage hike. I would really like to see results point to this, or arguments that could make sense of this possibility. However, my impression is that you won't have your free lunch and it will have unintended consequences. It's unlikely that making low skill labor costly will hurt wealthy citizens or people with degrees in engineering or law. If it does hurt people, it's going to be the people who do not have as much education or experience.
Another possibility that is seldom discussed is that you might be partly right. Maybe it's something like a thresholding effect: you have a window in which you can jerk the price of low skill labor up and get exactly what you have in mind at no cost. My intuition is that your best chance is that it's a window: at some point, the stories above will just become true for so many cases that you'll start seeing adverse effects. Much of what some people have in mind from ECON 101 is a very simplistic model; a more realistic version would have distributions of skills, background, firm types, etc. In that case, maybe you can get a window where the redistributive intuition works as intended. It's very hard to say without actually writing things down seriously into a model, however.
