“The GOV is also responsible for collecting taxes, SS, etc. Employers play a role in that too.”
Excellent, point! Employers have become unpaid tax collectors for the fedgov.
Have you ever employed anyone before? Do you know how much time, money and effort is required to collect state and fedgov taxes?
And it’s all done for free. Fedgov pays nothing for the service.
Now, employers have to become border guards as well.
For free.
Payroll services are actually pretty cheap. So is having a tax accountant. It's part of the cost of doing business.
OTOH, e-Verify doesn't cost anything. I'm going to guess it's not an expense or time issue, it's that you can hire illegal aliens much cheaper under the table than Americans. The GOV services they and their families consume can get passed on to other people, rather than they (and you) paying their fair share.