There is no good solution to this, and I think it's good that you've thought about this (sincerely), we need more people to be looking at these problems and thinking about them without ideological blinkers. My view is that I don't think it would work quite as well as you think. The arguments I've heard against this is that for government "employer of last resort" type work you:
1) get rid of any current low paid work covered by government - for example street cleaning and rubbish collection, since you don't need to employ people to do that work any more since you're getting the unemployed to do it - unfortunately probably the people who were until recently doing the street cleaning and rubbish collection, but now are unemployed. Worse they are taking home less money than they were before, are more dependent on welfare programmes and additionally are stigmatised for being unemployed.
2) (most of) the work they would do is useless - otherwise it would have been being done before. The people doing these "make work" are likely unskilled, so you can't get them to do anything requiring skills, and they probably don't want to do it (they are essentially being coerced to do the job) and will do the minimum they can get away with - for example look at community service sentences for offenders who have to pick up litter to "pay back to their community", they do an extremely poor job of it and need constant attention from an supervisor (who could do all the work on their own more quickly and more effectively).
It's not as simple as just being unproductive though, not only do you need someone to supervise that the unemployed person is actually doing something to "earn" their $10/hour, you need to keep records. So it results in a net loss of productivity - you are taking a productive person and now putting them on an unproductive task and you are employing bureaucrats to administer the system. If the cost of the bureaucracy is more than the worth of the work then it's a net loss - you might as well just pay people to sit at home and it would cost less no matter how galling it feels to pay someone for doing nothing.
3) Finally, there is the issue that if companies can get away will paying less per hour then they will do. It will mean that more people end up needing welfare support and overall it just ends up costing the government more in welfare and companies giving more profit to their shareholders - who if it was all fair and equitable would end up being the ones who pay the additional taxes so that the government has the additional money to pay for the additional welfare. We all know that that is not how the tax burden would be distributed though, it will be the middle classes who suffer under that additional burden. Ultimately, the cost of the bureaucracy of the welfare programmes can be more than the that of administering the minimum wage and the loss of potential increase in productivity through having those additional jobs from work that is currently unproductive at the minimum wage level and so ends up as a net loss to the nations wealth.
It's a fine balancing act and as I said originally there is no good solution that can solve all the issues. We just have to try and find the best balanced solution - some level of minimum wage and some safety net programmes - without allowing ideology cloud our judgement (on either side).