The proof is self-evident. There are a plethora of commercial labs that were not allowed to do testing, and the CDC's one blessed test was a failure. Had they not been the sole blessed entity allowed to do testing, states could've run things better on their own.
Better testing could've contained it long enough to allow us to prepare for the mitigation phase we're thrown into, unprepared again.
As for the states, not every state would've done well, but it's practically impossible for it to be handled worse than the way it was. The US response has been a disaster from the top down.