Comment by raattgift

No, not really. To boil it down to thinner text, and to focus on your "Space becomes timelike", I think you are stuck on (a) a particular system of coordinates that (b) are not regular across the horizon and (c) thinking that either of these does anything physical to free-falling infalling test particle.

The huge flashing red warning sign on (a) & (c) is that you drop in the words "'upward' direction", "{toward, closer to, away from} the singularity" and most especially "slower": you are clearly implicitly slicing spacetime into space and time.

If you can handle thicker text, Unruh has a nice discussion of regular systems of coordinates at http://theory.physics.ubc.ca/530-21/bh-coords2.pdf Additionally, Martel & Poisson 2001 <https://pubs.aip.org/aapt/ajp/article-abstract/69/4/476/1055...> (arXiv version <https://arxiv.org/abs/gr-qc/0001069>) is a nice discussion of PG coordinates.

More visually, one can compare the light cone structure on a KS diagram like at <https://tikz.net/relativity_kruskal_diagram/> (just before the "Edit and compile if you like") and a randomly chosen but very typical diagram in Schwarzschild coordinates <https://www.researchgate.net/profile/Ward-Vleeshouwers/publi...> or (in German) <https://yukterez.net/f/einstein.equations/files/schwarzschil...> (hovering over a diagram displays some light cones). Which cone appears to topple over in their respective coordinate charts is pretty obvious, and should give you plenty of shaded grey to think about the coordinate-dependence of "Space becomes timelike".