Furthermore, who is to say that because a job or task doesn't produce $10/hr in output that job or task isn't a perfectly valid and useful job?
People who are forced to live on $10/hr. The minimum wage ensures that people who work hard for a living aren't forced into poverty just because competition for work turns wages into a "crab bucket" situation where people undercut each other just to have *something*.
You can see this especially today in the wake of the 2007 financial collapse. Poor people used to be able to / have to work multiple jobs to make ends meet. They might end up working 50-60 hour weeks, but they could make ends meet. Today, however, with labor oversupplied and jobs in high demand, employers are able to demand "continuous availability." Not only will you not get a job if you want to have a specific day off or not to work on holidays, but you also can't get a job if you have another job that might want to schedule you at the same time. So people are forced to get by on single jobs that won't schedule them for full time and won't let them fill the hours at another job. You think minimum wage is somehow generous? Try living on it with only 30 hours a week. It's a large part of why the economy isn't recovering.
Besides, if a job is worth less than minimum wage, then it will be replaced with a solution that allows one person to do multiple people's jobs (aka increased productivity), or it will be brought up in value by charging more to the customers. Before you jump on the latter as making minimum wage hikes pointless, note that labor isn't the only factor in any goods' or services' costs, so the effect of raising wages isn't negated by raising the costs of goods and services, and this is a net benefit to the poor.
In other words, the minimum wage is possibly the single most important tool in ensuring that the income gap doesn't generate civil unrest and violence, like it did in the 19th century.