How would a “benefits entitlement” card even work, considering most schools are local or state funded and immigration is nationally controlled?

In most states outside of CA, school funding is largely from property taxes. Commercial property is taxed higher than residential, generally speaking. Therefore any employer that has a place of business is already paying more school tax.

As far as welfare beenfits, I could see denying them to non-citizens. But not unemployment or social security, if the guest worker has worked for the requisite time period and paid into the system.