I understand that. So get legal immigrants with work visas. That way you know who is in your country, you can keep track of them, and if they do break the law in a serious way you can revoke their visa and deport them
There are two reasons it is not done this way. First is that although immigrants on a work visa would not have to be paid a minimum wage they would still have to get some protection from outright exploitation. This would make it slightly more expensive for employers, cheaper than hiring American citizens but more expensive than illegals. And the second reason is that you would have to invest immigration services (start putting money into immigration courts, public defenders, etc). So it would cost, but it would help with the immigration problem, and it would just be the right ethical thing to do.
Everything you say makes sense except for one minor point..... Keeping track of them.
Couple of other points
1. Are you going to allow the families of those on a work visa enter the country also?
2. If yes to #1 are you as a taxpayer willing to support the education of the children
3. Do they, worker and family(if allowed in) have access to subsidized medical programs (say medicaid) or if a new health care act has subsidies would they qualify
4. Will a guest worker be allowed to collect Social Security benefits, or Medicare benefits? If no will they have to pay FICA taxes?
5. How long are you going to allow them to remain in the country if they are unemployed
6. If they are unemployed do they qualify for unemployment benefits?
Just a few issues with the guest worker program. I'm sure there are many more.