Yes, you are: the sheer amount of data that "backstops" this data. Part of the work is simply collating all of the information on each person into one pile.
Part of the process--and it would be a multistep one--would be a Social Security number check. A bogus SSN is enough to get you an extended stay at a Graybar Resort while everything else gets microscopically checked, subsequently to be deported to your native land. Also, the amount of data currently available on each of us LEGITIMATE residents of America, if it were collated into one usable file, would make identity theft a real challenge--not impossible, but a lot harder than you're describing. Among other things, assuming someone managed to steal a valid Social Security Number, cross-checking through the other data tracks would reveal the fraud pretty quickly.
Put the employers that hire illegals in the hot seat, just like the TAXPAYERS with the IRS…no matter how good their fraudulent IDs are. If an employer hires illegal aliens they pay a fine to such an extent that they will not want to continue the practice. The reality is that most people can recognize the profile of an illegal alien. A rigorous background check of an applicant should take care of any uncertainties. Sure, this will cream some businesses. SO WHAT! The IRS is creaming TAXPAYERS every year in part because of our lack of immigration control. Enough is enough.