The article you linked to already makes a good case for overall excess deaths vs. just reported cases, namely that doing so allows us to avoid both missed cases and those deaths that are only indirectly caused by the pandemic (lack of medical resources, people being afraid to visit the hospital).
Another thing that an "excess deaths" figure allows us to do is to preempt the argument that "most people would have died soon, anyway". If this was the case, we would see a spike in excess deaths, relatively quickly followed by a "death deficit" (for lack of a better term).
Come to think of it, we probably will see this deficit eventually - the question is just whether it will happen soon, or drawn out over the next years or decades. The more time goes by before all of this is evened out, the more overall person-hours will have been lost to this pandemic.
Of course, all of this is based on the idea that we can accurately predict what the baseline of "expected deaths" actually is - and as the article states, we probably can't do that in all regions of the earth.