Could anything take the mantle?

Yes and no. In theory anyone can get to the point necessary to claim it, but certain conditions must be met first. Initially Oryx's killer could simply claim it, but our rejection of the mantle has altered things. What specifically must be done remains unknown. Whatever it is, it seems a rather significant amount of Sword Logic must be acquired as all of the claimants so far have been pursuing that. To what end that Sword Logic is used for is unclear, though I think it is reasonable to think they may then have to kill us since we're the intended successor.

Something like Crota level or higher?

Meh. At least you brought a valid point