Except your logic falls apart pretty quickly.

You are choosing to blame people who are getting exploited. Either firstly they are being hired under the table to do this work under the market rate. Wouldn't this be the fault of the employer then. They are hiring such workers who are also not paying "their share". Conservatives are so quick to point at illegal immigrants as the problem but rarely the people who you know actually pay them to make living in this country possible for them.

Now what if they were stealing someone's identity to work a normal job? Its not right but identity theft is a large problem because we Americans for some fucking reason refuse to establish a national ID system. Where there are safeguards and ways to fix this but nah we are stuck with Social security numbers, which were designed merely for Social security for those above the age of 65 to claim their benefits but then it was misused by the IRS (which can we seen as fair since its a public entity in government) but then it was then abused by private organizations like banks and employers. Side tangent over, if an illegal immigrant did this they would then be paying "their share" of taxes. So your complaint is a nothing burger.

If we had an illegal immigration problem it is because there is a clear market for people who provide for them and it ain't only the democrats. Republican's Corpo rights focus is all about this and letting corps do whatever the fuck they want to do but focus on the suffering of others is their promise to their dimwitted voters (who vote primarily for abortion bans and immigration problems).