I wouldn't recommend
Banded Armor (550 gold)
Scaled Helm (550 gold)
As first items... You don't care about armor with Erebus bite, and purchasing monk idol is way better in my opinion as you'll beable to face 2 DG directly in the early game with those and bite.
So in priority, buy monks, and then take an helm giving mana (i take the one costing 1500G) it'll allow you to spam bite and mass charm to kill non stop.
+400 Health, +5 Health Per second, +525 Mana, +4 Mana Per second are reasonable increases early game. Your Bite is only a 300 heal and you don't start with a large amount of mana. By purchasing the Banded Armor and Scaled Helm first you increase your survivability early game. With that increase I've been able to taken on 2 Demigods with those items and Bite since I have higher survivability that most Demigods on the field at that point.
Another thing wrong in what you describe :
You advice the combo : bat swarm -> mass charm -> bite.
Here again that's not ok at all. You must keep mass charm to prevent the DG to teleport or if there are many others DG with the DG you wanna kill. If he is alone, you most of the time don't need to mass charm until he tries to teleport.
It's situational if a Demigod thinks he can take you. You're not going to want to wait until the last second to use Mass Charm especially if you have muddle to increase their cool downs by 100% while you fight them.
And another point is you don't recommend to take idols which in my opinion is here again an important thing with a general, even if you are assassin spec. Minautor and arche often take damage instead of you dyuring battle so it's a good way to avoid being focused by towers and all creeps at the time.
There is no point for Idols in my opinion if your going against a Demigod he isn't going to focus on your idols he's going to focus on you. Bat Swarm will instant kill the minotaurs, archers, priests, and just about finish the Catas leaving you the giants which will be hurt from the Bat Swarm. You shouldn't be taking much damage from creeps because of Bat Swarm and your Night Walkers that you can easily spawn and use to take damage from the creeps if necessary as well as the tower saving you mana from summoning Idols and money from purchasing Idols.
And finally : You NEED boot of speed !!!!! You absolutly need it to kill escaping DG, to move more freely around the map and all !!! (Unless you have others items with speed boost)
You don't need boots of speed. Erebus already has one of the highest base movement speed. He starts with a base movement speed of 6.3 the speed anklet favor item will give you 7.2 movement speed if you take it. Bite reduces movement speed by 30%. Bat Swarm has a range of 30 yards you'll be fine catching up to and escaping from Demigods. Save yourself a 1000 gold and increase your survivability.
Here is my suggestion as the items to pick in priority :
Idol monk
Plenor Battlecrown
Boot of speed
And there if you really need mana, take the crown at 1750G and then keep your money until you can take the crown at 6250G or so (the 25% mana reduction aura is the most important thing to be god like with Erebus)
And after, priorities again the boot a 6250G to have more speed
Then the level 4 monk idol and the others lvl 4 idols to have better damage deal and regen.
With those items, you are quite invincible in 1vs1 or 1vs2 because if you played well, you should have killed many others DG, having better stuff than them, having a better level than them and so, being godlike for them.
My two cents
With my suggest items you are quite invincible as well and it has been shown when playing with TheUnholyones and other players. Though there are other ways to play him. In my opinion the heal of the priests isn't necessary, but optional as they only heal 13% of your health when not buffed by Sedna. Besides your minions aren't going to last against a well played Torchbearer or other Demigods.
I see no mist. Mist is awesome as it turns you invulnerable for a short time. I've actually acheived quite a few kills thanks to mist (largely against slow moving characters like Rook)
Again Mist is one way you can play Erebus, but I don't use Mist when I use him. Sure it turns you invulnerable for a while, but I see no benefit to placing one point into it or a couple points into it. Yes, you can score kills from Mist depending on how low on health a character since and where you use Mist. Rook might be caught in it due to his slow movement, but a Rook that stacks on movement speed and uses the Wand of speed won't have much trouble passing by. Sedna won't have trouble, Unclean Beast won't have trouble, Oak won't have trouble. So it's all situational on how effective Mist is because it depends on the level of Mist and where you place your Mist and how you use it. Some people say it's good for escaping and I agree with them, but as I say many times if you stay aware of your surroundings there will be no reason to depend on Mist for escaping if you have Bat swarm and a watchful eye.