Why do you keep repeating that it has been debunked? MW increases put more money into the hands of people who will spend that money. The shoe maker will experience a sudden increase in demand and sales. What will he do? He'll order more shoes. (AKA more production)
I will concede that it's possible that a minimum wage increase could temporarily aid in stimulus on a macro level (though there are plenty of factors at play, so it may not and regardless, leads to long-term issues we continue to face from it), but it's definitely not as simple of a relationship as you make it out to be on the micro level. People having more money does not necessarily mean that all producers will benefit. Only some will, others may see their profit margins shrink, be forced to sell at higher prices, etc, while not seeing a significant enough up-peak in sales to be any better off.
Who benefits? Well, it is clearly the big conglomerates who are already eliminating costly labor with technology and outsourcing. So what if you make them overpay for a few button-pushers, if it means they gain a bigger share of the market, and can better set prices with less competition to drive prices down.
How does it eliminate competition from smaller companies and hurt them? Well, it's simple. Most smaller companies are operating on thin profit margins already, so they only have a few choices when wages rise:
1) Raise prices - certainly not what they want to do unless the supply/demand/cost curves determines that this is profit maximization, but they may not have a choice, because they must maintain a profit.
2) Do more with less employees - by either replacing them with technology (if they can afford it), or by hiring more productive employees at higher wages. Though employees can only do so much, and in many cases cannot compensate for the production of employees lost. You need to be able to pay labor at a reasoable rate for what they can produce for them. They have to.
Thus, it may not even be possible for most companies to overpay for labor, as it simply kills their profit margin no matter what. Small businesses are most vulnerable, and I assume that you realize that these people not making money is going to hurt, perhaps even moreso than the marginal rise in wages.
How will it hurt? Well, the obvious reason is if their business goes belly up rather than succeeding, you take their money out of the market. But also, as I mentioned earlier, less competition is worse for everyone but the few big companies who are actively eliminating employment in favor of technology.