My understanding is that liability is primary in those seven states and secondary in most of the remaining states. If a renter does not have existing liability coverage, then by definition and default, the rental company's secondary coverage becomes, in effect, primary.

The only state that explicitly allows the rental company to refuse to transfer their state-minimum fleet liability policy to renters is California. In all other states, the company's liability coverage does extend to the renter (on a primary basis in seven states and a secondary basis in the remaining states, a distinction without a difference for someone without their own liability). That is exactly what happened here: the OP does state that the claim was covered, whereas in California, Hertz could simply have denied to pay anything out at all and would have sent the OP's contact information to the party seeking a payout.

The real question this thread brings up, then, is whether the rental agency is allowed to pursue the renter to recover damages paid out by the company's third-party liability policy. That is an interesting question, and I am not sure of the answer to it.