No, that's not correct. The fact that the singularity is always in your future inside the horizon is an invariant, independent of any choice of coordinates.

> the singularity is still just a point in space

No, it's not. A point in space would be a timelike line in spacetime. But the singularity is a spacelike line in spacetime. That's a moment of time, not a place in space.

> There's no need for this whole "space turns into time" notion

That's true; that notion is an artifact of Schwarzschild coordinates. But that does not imply the other claims you are making.

> the spatial coordinates of the singularity

I'm not sure what you mean by this. It's true that, since the singularity is a spacelike line, you can treat a coordinate that varies along it as a "spatial" coordinate marking different spatial points on the singularity. But the singularity itself is a moment of time (as above, a spacelike line), so it is not a "place", and it does not have a particular set of "spatial coordinates". A spatial coordinate marking different points along the singularity is marking different points in space at a moment of time.