Is it possible, that the demon - a modified fury - can only attack every 3 rounds, because his only available attack option is this flyby thing which a recharge rating of 3? I couldn't find any other attack actions for him.

Ok, he could do the chaos mumbling thing, but I had do let him fly over the pcs for 2 rounds before attacking again. He was good, but not especially awful, eventually he was shot from the sky. A bit anticlimactic, my group only had a sense of panic, because two of them got unconcscious through the terror effect after he appeared.

i used "Melee basic attack". All monsters should have access to a basic attack, if not listed in their info sheet, just use the Basic action card i'd say

All monsters have access to all the Basic action cards, such as Dodge, Parry, Melee Strike, Ranged Strike, etc.

The GM can also give NPCs access to other action cards, at their discretion.