I do not believe that is accurate. The reason that out of state drivers are exempt is that these new fines are not enforced by the Courts (where one would have due process and other rights). They are tacked onto your license, and you have to pay them to renew your license. My understanding is that the enforcement is done through DMV licensure proceedings, so there is no basis to "catch" out of state drivers in this net. The disparate treatment of out of state drivers is really what has most people riled up. As I understand it, going through DMV, rather than the courts, also severely curtailed the drivers' ability to challenge the fines. Bottom line, I think, is that the legislature decided to take enforcement shortcuts to grab this money from an unsympathetic group-- bad drivers-- and didn't count on the backlash from the disparate treatment of Virginia and non-Virginia drivers.
You’re right, Kaine did not line item veto out-of-state drivers.
I can think of only a couple of ways to make the people who use the roads pay for them — the gas tax (really a road user fee) and toll booths.