Youth unemployment - well I have a lot of opinions about this - and your answer is correct - you up and leave and chase the work, you don't sit around and complain. Part of the problem is that we tell everyone that they're going to have a wonderful career when that is plainly not an option for every school leaver. What should be taught is self reliance and basic financial literacy and let people make their own decisions from there.
As for that unemployed capital - there's plenty of scope to put it to work, but the regulatory environment closes off a lot of options, minimum wages being just one of the factors. It just takes some creative destruction and some new ideas to get it moving again. I don't have any of those ideas (nor any of the capital, for that matter) but I'm sure there are plenty of new things just around the corner.
Where we differ is about where the benefits go. Of course, we can all benefit as customers from cheaper books or more reliable vehicles and so forth; indeed, Walmart exemplifies the benefits of scale and modern supply chains in a way that substantially lowers the cost of living for a very large number of people. But it is not true that all of society benefits evenly when dull jobs are eliminated by automation.
Sure, those warehouse workers never get that job, but there are jobs at the robot makers, robot technicians, systems designers. The freeing up of this labour enables it to be deployed elsewhere, making everyones lives better.
See, if we had low unemployment and steady growth I'd agree, because industrial policy props up inefficient businesses and labor shortages drive up wages and act as a brake on growth. But what we have now is unusually high unemployment following a long period of wage stagnation, even though we had a period of record GDP growth at the same time. Between technology, outsourcing, and a shift towards services, it's not clear where less skilled labor can be efficiently deployed in the US at the moment. Youth unemployment is something like 40% because there's just not much demand for the inexperienced. One approach might be to abolish the minimum wage so that people could be busy doing something, even if only for $1/hour - but that's not a realistic policy, and would probably lead to riots.
It's no good being able to deploy your labor elsewhere if you lack skills that are in demand or the place demanding them is far away. Of course, you can up and move. I've done it myself, on several occasions - but I was young, free and single, and more importantly I had portable skills that were in high demand. The sad fact is that a lot of people aren't very entrepreneurial or smart - half the population is on the left side of the IQ distribution. A cheerful distribution and a good work ethic are important for anyone, but absent the skills most in demand, or the knowledge of what those even are, the opportunities are limited. We can easily come up with suggestions for a given individual but it's much harder to do across a population. And part of the reason for this is that neither government nor the economy as a whole operates like a business. We don't have the option of just firing 10 or 20% of the least productive citizens.
There's plenty of capital sitting unused in the private sector right now, but not enough new demand to invest it in new equipment or hiring. The ~30 million unemployed and the bottom 50% of earners don't have enough money to do much shopping, so it's not going to come from them. Europe and Japan are in a similar situation. Where is the demand going to come from?