1) American businesses are not allowed to hire those who are not allowed to work in this country. Some do with farm or construction work and if they are caught the fines and penalties apply both to the business and the individual. Having a verifiable SSN is a protection to the employer whether it is day labor, independent contractor or employee.
2) Providing a verifiable SSN allows the MSC the assurance that you are you and not a person they deactivated already for cause.
3) Providing a verifiable SSN allows the MSC to have some measure of confidence that you aren't registered with them multiple times to avoid rotation requirements on shops.
4) Providing a verifiable SSN allows the MSC to have some measure of confidence that you aren't re-registering over and over in order to keep your annual earnings below the level at which a 1099 tax report.
Identity theft? Yes, there is a risk. But there is a risk with every single job application you put in, bank account you open, insurance policy you own, doctor's office you visit, credit card you apply for, cell phone contract you sign and on and on and on. At the present time I am being offered identity theft protection by the Department of Homeland Security because I was named as a reference on an application by a friend to work for the TSA because Homeland Security was hacked. I am offered identity theft protection by the facility that does my occasional bone density scans because the dumpster carrying their old paper records to secure destruction opened in traffic and my records and those of thousands of others were blown all over the roadway. I am offered identity theft by a retailer whose system got hacked. How many other places have been hacked or thrown away papers with my SSN or other personal data and have never even acknowledged it?