I agree with you that employers must pay at least the minimum wage to its employee's, and this should be true regardless of legal status of the workers, and I also agree with you that teenagers and young adults today are just to be paid the minimum wage. Actually, this is already true in our economy with individuals under 25 making up for around half of all minimum wage workers. It would be unreasonable to pay, for example a farm worker significantly over the minimum wage where their job entails no specific training or education.

But I also believe that growers today enforce a sort of plantation style mindset where they prefer to hire large amounts of illegal immigrants over investing in mechanization and other methods that would lower their labor elasticity. While I cant criticize them for how they run their business, since it makes sense to keep labor costs low instead of investigating in machinery. The reason I stated Martin Luther King Jr. was because there are already various examples of human right abuses by growers. Since illegal labor is dispensable and easily recycled, it paves the way for growers to harass and mistreat laborers, making them feel superior, while the laborer inferior. HRW posted a very good article about this if your interested in reading it.
https://www.hrw.org/news/2012/05/15/us-sexual-violence-harassment-immigrant-farmworkers

A good step the government should take in my opinion is introducing subsidizes to these growers so they can raise their efficiency and stop the steady stream of immigrants they hire. As you said today's policies are inefficient, maybe by cutting the demand of immigrants by technology instead of raising wages to attract domestic workers would be a better solution.