People believe this because every single member state is using EU institutions as a punching bag whenever they have issues locally. The people have no idea how the EU work, they only hear about it when used as a bogeyman
It also ignores the fact that absent the EU, countries would still have a lot of subsidies.
Edit: I just now got the part "If the increase in cake size is bigger than the subsidies" – that's a ridiculous assumption. EU total growth in 2025 was 1.5% or 295 billion USD. From 2021-2027, the EU budget committed roughly 370bn € each for Agriculture subsidies and for Cohesion Policies, totaling 720bn € over 7 years; 2025 has seen ~130bn USD from those two buckets alone. Germany alone has paid another 60bn in subsidies the same year. Oh, and there has been a total of 417bn USD in energy subsidies in europe in 2023 (most recent data available). Even if we could attribute all total growth to subsidies alone, we'd have a factor below 0.5 – and that doesn't include any growth through private investmen, PPP, or any kind of increase of regular public spending.