It may be somewhat counterintuitive, but it's certainly not illogical.
> Minimum wage laws are incredibly damaging as well. For example, the unpaid part-time "job" that launched me in the IT field would be illegal now. And I would be much poorer for it. But, hey, I wouldn't have been "exploited".
First of all, are you sure it wouldn't have existed? Just because you're willing to get something for free doesn't mean you're not also willing to pay for it; and minimum wage is a lot cheaper than a properly trained full-time IT person.
Secondly, having skilled workers available is a public good [1] that benefits all companies, so it makes economic sense for the government to invest (on behalf of all companies) in developing those skills. If these sorts of apprenticeships are generally useful, then having the government sponsor or subsidize them also makes sense.