You will need someone a little more experienced than me for ideas of implementing stormcast. I can answer a few things however.
Not running stormcast as general is no big loss really. With my CoS list I'd usually run a CoS general anyway as the different generals allow for different units to become battleline.. So you can just pay less battleline tax really.. Which is good. What generals unlock what units to battleline is usually dependant on there old WHFB armies they used to be in before the merger. So Dwarf general allows dwarf hamerers to be battleline for instance. Your general on griffin would allow swordsmen to be battleline as a specific answer. There's no stormcast general that moves other stormcast to battleline in CoS anyway. So no loss there.
One exception is that Aventis Firestrike can be general in Hammerhal..you get an extra CP once per game for this..
Which is underwhelming as now you can't take the command trait (unique character).. But he's the only exception.
The only stormcast I run in CoS are units that do anything my CoS units can't. Knight Incanter (free dispel) and Knight Azuros (re roll ones) are two good examples.
Hope this helps.