I think what you wrote is accurate. Contracts I’ve written over the past few years include clauses that the supplier/contractor will ensure that everyone that works for them has a legal right to work in the US, they will use e-verify or whatever. However, it wasn’t that long ago that we never thought of using such language. Further, I haven’t seen many construction contractor agreements in a few years, but the ones I’ve see use a standard template and it’s a bit difficult to negotiate any clauses (though Trump certainly has more clout than I did).
True, standard AIA contract forms have been developed and court tested over the years to become defacto industry standard. I’ve been out of the business since before the e-verify existed, so I’m fairly certain it has been incorporated into their documents.