I don't think it works like that though. They already have incentive to create higher productivity jobs, and that's making more money. The more productive they are, the more money they'll make. But increasing productivity and creating high productivity jobs doesn't happen overnight. For example, how could a business get started in the first place? Raising minimum wage actually favors the big greedy corporations and hurts the small businesses. It wipes out the competition from smaller businesses and businesses that are just starting.
Take for example, Walmart. They're one of the biggest businesses in the US, and employ the most people. Minimum wage goes up, they can afford it, because they already have enough capital to pay the new minimum wage without taking a significant hit to their profits. Smaller retail providers (aka competition to Walmart) with less capital will take a bigger hit, some may go out of business, and it will be harder for new retail providers to start up unless they already have a significant amount of capital to start with. Walmart just got a lot less competition.
That might sound like a bit of a stretch. But still, consider what it will do to the smaller businesses?
You're right about higher productivity jobs not emerging overnight. However, any kind of transition takes time and if we want to move forwards we will have to deal with short term costs, such as potential layoffs (though as I addressed previously it's not as bad as theorized) and unemployment, but such costs would be inherent to any kind of change. There's no free lunch.
I would argue that worker productivity has no direct correlation to increased profit. If you think about e-businesses like Amazon and eBay, designing better systems and websites would do more for their bottom line than increase productivity at head office. Companies with a strong incentive to increase worker productivity are those who need to invest in their human capital. On the other hand, I don't think large corporations have an incentive to create higher productivity jobs - jobs like those at Walmart exist because in such an economy of scale you're able to create a division of labour so fine that you're a human drone. I would say for these kinds of corporations, their model disincentivizes creating higher productivity jobs, they don't need you to do good work and doing good work (as a Walmart worker) doesn't really help them. For this reason, most minimum wage workers work for said large corporations. Small business tend to pay higher than minimum wage and I'd imagine that the work is better.
This is what fast-food managers did in response to raises in the minimum wage in Georgia and Alabama. Furthermore, this is what they did in terms of increasing efficiency:
I think there is little evidence that increasing minimum wage would reduce employment and increase pressure on small businesses. I've worked for a smaller business before, and there was much more pressure on developing our ability to sell than workers in a larger business in the same industry (I've done mystery shopping as part of my work so I know we put a greater emphasis on selling :D). In a small business, workers worth relatively more (because as you mention there is less capital), and as a result they would do more work and be in turn paid appropriately. Like I've said before, most near-minimum wage workers work for large corporations, as smaller corporations have "better" (more complex) jobs and pay more in order to compete with large corporations for labour (as everybody thinks about applying to Walmart or American Eagle rather than your local electronics or clothing retailer). So, we can force businesses to be more efficient, which makes sense. Firing people is hard. And there's an alternative to doing so.
Furthermore, to expand on the increase in demand (which is the other main reason we should increase the minimum wage in addition to increasing productivity). The minimum wage would obviously affect low-income earners the most, and their increased wages will mostly go into increasing their spending power. Poorer people spend more money relative to saving it. While there's always fear that the money put into consumers hands won't be spent (which is the purpose), increasing the wages of the poor is more likely to translate into spending power than for any other group. It's something else I've learned at work (which catered more to poorer folks), that many poorer people don't save money - don't even have bank accounts.
I admit that I don't have much training in classical economics. However, my interest is in political economy and development. I've come to appreciate the role of the state in creating change in society. Crucial to that is improving skills and productivity. I believe that mission should be pursued by all actors in society, and if certain actors don't have an interest in doing so (like Walmart for example), we should give them incentives (to put it politely)/push them (or not) to do so. Regardless of whether we agree on the effects of this policy versus the next, we would agree on that the status quo must change, yes?
To put it simply, I don't think we should fear an increase in the minimum wage. We should not. Other countries have higher minimum (and average) wages than we do, and if we're concerned about American prosperity for the masses in addition to American prosperity in aggregate (which mostly benefits the wealthy or educated), then I think we can notice something is wrong when the median worker is paid so little for all the boasting we do about our wealth. That must change, doesn't it? And if the government won't lead the way, who will?