You are correct that I oversimplified a complex situation. https://euromomo.eu/how-it-works/methods/ describes their method, especially section "Hypothesis".
Their model includes a sinusoidal seasonal component, as you noted. However, what I said is also true. From the Hypothesis section, third bullet:
"Parts of Spring and Autumn are less likely to be influenced by additional processes leading to excess deaths and the main pattern of mortality can therefore be modelling using only those periods, resulting in a baseline being the number of deaths expected when no particular process increases mortality."
As is clear from the bullet that precedes the text that I quoted, "processes leading to excess deaths" means winter infections and summer heat waves. So my description, while incomplete, is hopefully not too misleading. For completeness, I should mention that they also adjust for population growth. (You can't just do 2020-avg(2015-2019) because the total population is different.)