There's where you fail to follow it through.

Businesses that engage in illegally employing illegals will be punished, harshly.    Welfare denied to illegals.   Carrot eliminated.

If a business needs to hire an American worker, they'll have to compete with other companies for that resource by offering higher wages and better benefits.

See?    Now it sorted itself out and you didn't need to introduce silly minimum wages - capitalism just fixed the problem and everyone's happy.