In principle, not a bad argument. But, the practical issue is that the government already requires some burdensome documentation. Both my children were recently hired for part-time jobs. The level of documentation was considerable, and the paperwork quite burdensome. For example, only certain pieces of documentation were sufficient by themselves. The only single piece of documentation that I had on my own children without going to the safety deposit box at the bank was their passports. Otherwise you needed two. Therefore, the burden already exists.
It doesn't seem to work though. So, replacing with an instant check system would reduce the burden and improve the effectiveness.
This is the identical argument the NRA made with regards to background checks. The NRA was not opposed to background checks for firearms purchases, they just wanted it easy and fast. (Previously, it had been designed deliberately to take weeks and be very burdensome. Now it takes just a few minutes and you hardly notice it.)
If the government cracks down too hard on these guys, businesses will just refuse to hire Hispanics altogether, legal or illegal.
I never mentioned Hispanics. And indeed, the existing background check is done on everyone. My children are basically WASP / white, and they still had to do it. Indeed, the instant check actually protects businesses, which is why gun dealers love it. If the check says that the person is OK, they are legally not liable. Same for businesses.
If the check can be done in two minutes, and relieves the business owner of liability, then it would be a good thing.
According to Senator Cornyn (R-Tx.), my Senator, it would take five years to build such a database.