Depends on the exact properties of the core, but the hardest ones exhibit very low permeability below the coercive limit (basically, the self-magnetization is so strong that they remain saturated), so yes you would have an inductor just like ever, except it would be air cored. So you'll need a lot of turns. But you'll need a lot of amp-turns to push such a hard material into a different magnetization state anyway, so that's fine.
The trouble with magnets as cores is, they don't do anything until you exceed the coercive limit, and only then do they start storing flux. Until it saturates in the opposite direction (which is how you know it's fully magnetized in that direction), then it does nothing for a while again. This is a huge hysteresis loop, and it only saps power. So, it's utterly useless for power control applications; the energy density also means just one cycle heats up the material significantly, so you can't use it at AC anyway.
But you're probably on the right track. Saturation itself can be useful: if you can control the flux into a core, you can control when it saturates, and use that to transmit (or not) power. Such materials are used for magnetic amplifiers and saturable reactors. They are defined by a small hysteresis loop (low losses), high permeability, sharp saturation. The loop has high remenance (it remains permanently magnetized very near saturation), but low coercivity (so it can be very easily pushed away from saturation).
Examples include square ferrite (a few special kinds: not something you're going to find in generic ferrite beads or whatever), amorphous/nanocrystalline (most types), and permalloy and related (traditional rolled strip) alloys.
Such materials are generally made to exhibit as high permeability as possible (in the linear range), so that energy storage is intentionally minimized. For energy storage, the ideal inductor has a permeability of 1, i.e., no core at all (a core is normally used only because we don't have anything more conductive than copper).
Tim