Well, time pointing in a spatial direction is a non-sequitur. At any point in a manifold there is a light cone defining time like directions, light like directions, and space like directions. Any small region looks just like Minkowski space, including a region where the horizon is passing by at c. There is nothing spatial about a timelike direction inside an event horizon except that it is labeled r in some coordinate schemes. It's labeled U or V in Kruskal (depending on your convention). It's labeled t in local Fermi-Normal coordinates. I think it is genuinely misleading to attach significance to a letter used in interior Schwarzschild coordinates for what essentially are historic reasons.