The conclusion of that paper:

The higher the vaccination rate in a state, the lower the deaths after the vaccine was introduced. This is unaffected by the age of the state.

The fact that this shift only emerges after the introduction of the vaccine means we can pretty much rule out any demographics-based causes. The "low vax" and "high vax" states were pretty much even before the vaccine. The only reasonable conclusion is that high vax rates directly caused lower death rates.