> That's a strange requirement considering the executive of most EU states is not directly elected by people either

At least, it's usually the leader of the party the people voted for in the legislative elections.

In the EU there was this Spitzenkandidat idea floating around ten years ago, but it was never enacted in texts and died at the first opportunity (naming Von der Leyen back in 2019 when she wasn't the leader of the PPE), because the heads of members states (particularly the French) weren't willing to give up their designation power.

In practice there isn't even European political parties, the European elections are just national elections represented by national parties and most citizens don't even know the names of the European coalition of parties (PSOE, PPE, Renew, etc…).