The EU doesn't implement any of its policies by itself, ergo it should not require an executive branch of its own. There is no EU army, no EU police. We already have an instance of that in each member state which is required to implement EU laws on its territory, the Council of Ministers coordinates on that afaik.

The general process is a bit like this, simplified:

- the Council of heads of state appoints the Commission

- the Commission proposes laws

- the Parliament approves laws

- the Council of ministers implements them

- the Court blocks any unconstitutional laws

The problem has been for the longest time that the Commission appointments are not elected, somewhat mired in cronyism, and they keep proposing nonsense laws while the elected parliament can just stand there and vote no while not being able to suggest any legislation we actually need.