First off, you hit the other item right off. Having a minimum wage means that any task, or worker not expected(on whatever horizon they're looking at) to perform at or above the amount of pay required for hiring them, will simply be dropped/contracted out, or simply not done. So the 16 year old looking for a summer job with an announced plan for a 2 week family vacation in the middle of July suddenly becomes a lot less employable at $8.75/hour than he would have been at say, $5.00/hr.
Another effect is that lots of workers officially are contractors or subcontractors and not employees. Very few taxi drivers or whatever the newest car for hire firms are now turn out to be "wage slaves." Minimum wages that are not affordable, or that can be beaten by making drivers contractors who hire the vehicle from the cab company and "work for themselves" are a better bargain for the owners.
That hold through to the big trucks as well, in the tractor/trailer industry, the most expensive part of the operation is the truck, because you have the driver you need to pay, and then the operation and maintenance of the truck itself, and with the emissions requirements on newer trucks adding tens of thousands to the operating cost of the truck over its service life(cradle to grave). Getting drivers to step up and go "owner/operator" often becomes a lucrative deal for the Trucking Companies, and that is without certain ones allegedly pursuing predatory practices towards their drivers when it comes to trucks nearing the end of their lease and doing things to drivers to force a default. (Impossible load assignments, followed by "sorry, can't find anything for you at this time")
But yeah, minimum wage is popular with two groups. It's popular with the people who would see a pay hike with the change. And it is popular with the upper tiers of the pay scale because it makes them feel better, but otherwise doesn't have much direct impact on them personally.
That and certain groups have learned to play the emotion card on it for the people who fall in between those two groups. "Don't you care about the people who are only capable of earning ___ an hour as the minimum? Here let me share you a story of one such worker..."
Yes it sucks for the people who get trapped in that situation, but they've historically been the exception, not the rule. At least until we started actively distorting just about every market with well intentioned laws, from rent controls(which most areas quickly discovered was a disaster and eventually dropped), to housing assistance(as the alternative to rent control, but creating distortions all the same, as it effectively set a artificial "bottom" for the market, to food stamps(again, artificial bottom on the market), and so on with minimum wage being just another in a long list of them.
I do agree that between minimum wage laws and labor unions, those two things were most decisive factors in a lot of industry being shipped overseas. Environmental regulation may have forced some others, but I think those were more edge cases. It ultimately was the price of labor(and resistance to automation in the case of the labor unions) that decided things for those industries.
That isn't to say all distortions are bad, but that doesn't mean they're not distortions of the marketplace all the same.