I'm clearly no doctor or expert - I wouldn't know, just asking the question.
Countries have been doing it. Germany, for a start, though they did it on the basis of only counting deaths of people with no serious underlying conditions, I believe.
Also think they changed their tact on that in the last 10 days or so.
Sort of don't need to be a doctor for that mate - it's just common sense.
Say a patient deteriorates due to COVID-19 but ultimately it's an in theory totally unrelated heart attack that "pulls the trigger" on the death rather than asphyxiation, because of stress on the body.
In that case, you'd have to mark the death as "with" the virus instead of "from", because technically the heart attack killed him.
Simply wouldn't work and would make the stats useless.