The math in that article is based in Cartesian space. Note specifically figure 4, where a portion of a Cartesian plane is pinched in at one side to show the supposed warping due to gravity. Using the shown transformation, she concludes that space is compressed near a black hole rather than stretched. The diagrams after that along with the process explanations clarify this.
The "pinching" transformation used, though, is not accurate to how gravity warps spacetime. Note that the horizontal dimension becomes radial and the vertical dimension becomes circular. This is an attempt to transform from a Cartesian coordinate (length x length) into a polar system (distance x angle) but the math used afterward is still Cartesian math, which is incompatible with the new implied coordinate system.
Referencing figure 4b, I note that the wedge-shaped grid is about 30 degrees of a circle, so theoretically (in the author's system) twelve such connected grids would form a full circle. Assuming units in meters, the grid is about 30m wide by 10m high, so the final circle is about 30m in radius and 120m circumference at any distance from center. In the real world, a circle with larger radius has larger circumference. In the author's contrived system, orbital speed/period is independent of altitude. The math being used is not consistent with reality because the it is performed in the wrong coordinate system.
Another false premise at work here is that she uses a finite (30x10 in this case) space to illustrate warping of an infinite space. The only way to apply the shown transformation to an infinite grid would be to have a black hole that is in an extra dimension, I.e. "off the grid." Since black holes exist in the same dimensions we do, the math the author uses again does not agree with physical reality.
The article you've linked attempts to disprove a physical process in a universe with a particular geometry by comparing it to a functionally different process in a completely different geometry. The math stated in the article simply does not apply to spaghettification and thus cannot disprove it.