Chase provides cardholders with a booklet which explains the coverage that they do provide, and it doesn't matter if the cardholder thinks it should be different. Primary refers to the claim paying precedence, not to the quality of the coverage. Your personal insurance may provide better coverage than Chase's coverage, and you may need to use it if you cannot persuade Chase's carrier to process the claim because you had no prior knowledge of the incident.

Does Budget have good evidence that this damage occurred during your rental? Have you asked Budget's agent to give you the cite for the specific state law that allows them three years to notify you of damage?