In fact, you would be surprised at the number of doors that close if you don't have ID or have that crappy ITIN. EG: while I was studying, my friend only had an ITIN on his first year, and it sucked really bad for him. No access to credit, no access to most rentals (because no credit check to run!), etc. Even stuff was more expensive, as I could get the 10% Target discount and he couldn't.
(I was lucky enough to had a SSN from when I worked with a sort-of-diplomatic G4 visa, and it was incredibly useful, and allowed me to get a good credit score, a nice apt close to campus, etc.)
I understand that you are annoyed by having to hand over information about your kids to the govt, and maybe the process of getting them a SSN is a chore. But believe me, both legal and illegal immigrants would love to have SSNs.