Do you think Eb will choose Harry? I suspect he might choose someone else like Carlos, setting up a showdown between Harry, and Carlos.
If it didn't go to Harry, I would like to see it go either get claimed by Mother Winter (in which she then might get more active...) or be claimed by someone like Cowl. I think it wouldn't be interesting for Carlos to get it, and as others point out he isn't quite senior enough. A new threat, perhaps even a neutral one, might be interesting. A new character that isn't on the Council or the bad guys or even Harry's team.
In terms of Council candidates there are hardly any. Morgan would have been the obvious choice, I think. But now most of the old guard are dead or debilitated. Luccio could use a boost but considering how easy it was to compromise her I don't think Eb would pick her. I don't see him picking anyone else on the Council either, most because they either would become too powerful or because it would be a terrible burden to give to one of his friends.
Harry still seems the best and obvious choice of course, as it gives him another way of standing up to both the Council and many otherwise much more scary foes. But we'll see. Jim isn't always predictable. And he normally if Harry gets a power up it's either half-useless without causing more pain to regular people OR it just means Harry has an even worse time. So if Harry does get the Blackstaff I don't expect Harry's life to get easier. Jim also might choose to give it to another character, perhaps an antagonist of Harry's, because it allows him to create more story problems and makes Dresden's life worse. He does love doing that.
Eb has the journals though. There may be details of a Way into Demonreach's vault or a way to access the vaults remotely hidden deep in there. (Unlikely, but possible).
That said, if you were to put the Eye on the Blackstaff it would probably help protect the wearer from the awful side effects of a hate-based radiation machine (or it could draw on Mother Winter's which may be worse?)
As interesting as that sounds, it might just kill Dresden to attempt to use the Eye. It's not just the corruption effect. It's also the sheer power. It would be like handling nuclear energy.