CMIIW, I believe the doctors were already doing most of the work, and they were already ruling out non-covid so the worst this can do is allow a few non covid deaths to be reported as covid deaths.
I'm pretty sure all deaths were going through the medical examiner's office, and I'm pretty sure they were conducting post-mortem Covid tests in certain cases. My knowledge here isn't very strong, but my understanding is that it is very rare to have Covid as the only cause of death. It can be either the primary or secondary cause, but there are almost always other causes (comorbidities), often even caused by the virus itself, such as heart or liver failure.
When you have everything going through a central office, there is uniformity in how these things get documented. For example, someone who had a pre-existing heart issue who contracted Covid, that office had a standard for what they put on the certificate. Now, each doctor gets to determine if Covid played a part, how much of a part it played, and whether it deserves to be listed on the certificate. You can have the same type of case present itself 50 times and have half counted as Covid and half not. Without having uniform reporting standards, you have no idea what you're looking at when analyzing the data.