Instead of a lottery or affirmative action -- why don't we let everyone have higher education who wants it.
This country progressed with the GI bill and letting a lot of people who wouldn't otherwise afford college go. And we could do the same for trade schools. You only have to pay for education if you leave the country.
It's the reverse of the "H1-B Visa" where businesses don't pay for education in the USA and import people who got educations paid for by their countries.