That there is a 100% excess mortality peak in April tells me nothing if the
overall excess mortality is around 5%. As I say, it may be due to different factors, among other things that at that time COVID was something new. If you add another disease to the mortality rate we already had, even if statistically it has a low mortality, the overall mortality rate will normally increase. This is fully compatible with COVID cases being inflated, as has been done, by counting completely asymptomatic people as COVID or also counting common flu cases as COVID.
But from your link (my bold):
Relative excess mortality in the countries we have examined ranges from -4.3% to 14.4% and is strongly positively correlated with the recorded number of COVID-19 deaths (r = 0.8 )
A correlation value of 0.8 is pretty conclusive.