It's much less clear cut than that. In fact death with COVID ≠ death from COVID every single time, even for elderly people.
Therefore it's highly likely that for a given death it is more likely to be incorrectly categorise as non-COVID when COVID was responsible, than to be incorrectly designated COVID.
https://www.washingtonpost.com/world/britain-says-it-overcou...
Here is some data here on excess deaths: https://www.economist.com/graphic-detail/2020/07/15/tracking...