What we've learned from the last few years is that increased profits don't translate to more hiring. So no. Unless Americans want to work for the same wages as Mexicans, changes in trade laws wouldn't automatically mean more American jobs. And don't assume that they're living at home because they're unemployed.
The reality is it takes fewer workers to do the same job. For the past 20 years, the productivity numbers have gone through the roof. Those who have jobs are willing to work more hours, including salaried workers who don't get paid overtime. Automation and technology has eliminated lots of jobs. We see it in radio. Most of the job losses in radio have come from improvements in technology. The job is still being done, with less people involved. This is partly why one of the most popular courses of study in college today is entrepreneurship. Young people prefer to start their own small businesses, rather than work for a large company.
Those increased profits are partly due to cheap offshore labor costs. And decreased regulations costs. Which involve US Government policy, as well as the more corporate and business friendly government policies of foreign governments.
Sure, automation is part of the unemployment picture.
But there are probably hundreds of thousands, if not millions, of people employed in offshore factories owned by US companies, shipping American branded products back to the U.S. I'm certain that there is a lot of automation in those factories (especially in the Mexican ones -- I've seen photos of the operations in the newspapers) -- but it still amounts to a lot of lost American jobs. There are a lot of people employed in those factories.
The only reason Americans would have to work for the same wages as Mexicans to get those jobs back is due to Government trade policy. Sixty years ago labor in Mexico was even cheaper than it is today. But you didn't see American companies moving entire factories to Mexico the way they did after NAFTA (and even before NAFTA, due to previous trade agreements). There is a reason for that. Government trade and tax policy.
Eliminating or restructuring American job killing Government trade agreements would not be a panacea for workers. But there would be millions less unemployed in the US, automation or not.
As for your statement that "young people prefer to start their own small businesses", I doubt that. A small percentage probably do, especially those who are involved in the software business, but if you look at statistics, the majority of younger people in the US who are employed work for someone else. And if you also look at statistics, unemployment among the youngest working-age demos is very, very high.
Either way, Government trade, tax and labor policy either helps or hinders working Americans. It can help or hinder American employment. It can help or hinder US businesses.