Hegemony is hurt the most by the current imbalances with the vassal system because the vassal system does almost everything that a hegemony does, only much better.
If I were spit balling ideas.
-Vassals should eat a diplomatic weight malus, since everyone in a federation, the GALCOM and the Galactic Empire would be aware that they aren't independent. This would be a nerf to vassal spam, but would also help hegemons because that means independent empires in their federation would be more useful to getting resolutions passed than having vassals. I want to say, there is a federation law where you can force your fellow members to vote with you or maybe it was that you got a chunk of their diplomatic weight.
-They need to introduce federation holdings. If I were doing it, The bonuses for holding would scale with federation level. not sure if all four slots would be available out of the gate or locked behind the federation levels. For each vassal you have, you'd eat a malus that lowers the benefit you get from federation holdings., You're Hegemons and Galactic Unions would have access to all the federation holdings as a thing that sets them apart from other federation types. Then an area where those two would differ, is that a hegemon leader could determine what holding buildings are available or that other members have to build. So if there is a holding that boosts food production and you think it's a huge waste of resources, you as the hegemon leader could forbid members of your federation from building it, while mandating that at least one of their holding slots has to be filled with the joint federation research building.
-Even if they rework the cohesion mechanic to be less brutal, if it still remains, maybe vassals have a modifier that increases the cohesion they cause or have it so they are less able to mitigate cohesion issues within the federation. This would be more about making it less appealing to create a federation that is is almost nothing but vassals. I want to say, they did make it so that you can't create a federation with a vassal, not sure if they also made it so that your federation goes poof if all the other members are your vassals. Regardless, if it's the later, it really doesn't stop a player from creating a one system empire to be the sole independent AI empire in a player's federation, while all the other empires are vassals of the player.
-They probably do need to accept some immersion loss and give players more control over federation they are in, while the AI has less control because god is it dumb at times. I admit that would weaken the point of Hegemons, but making the AI less obnoxious to deal with, isn't a strong selling point and is kind of damn in it's own right. "Hey, your choices are have some really nice bonuses, that the idiot AI ruins or have okay bonuses, but the AI can't ruin those."