When reading this section it feels like there's a giant elephant in the room that is not addressed. For example, here's a quote from the section:
In 1920, Paul Ehrenfest showed that if there is only one time dimension and greater than three spatial dimensions, the orbit of a planet about its Sun cannot remain stable. The same is true of a star's orbit around the center of its galaxy. Ehrenfest also showed that if there are an even number of spatial dimensions, then the different parts of a wave impulse will travel at different speeds. If there are $5+2k$ spatial dimensions, where $k$ is a whole number, then wave impulses become distorted. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only with three dimensions of space and one of time. Finally, Tangherlini showed in 1963 that when there are more than three spatial dimensions, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.
Okay, but this assumes that in the hypothetical universe with only one time dimension and >3 spatial dimensions, the laws of physics as we know them remain valid. How do we know they remain valid? After all, if these theories were formulated in 3+1 dimensions, it can't be surprising then that they work best in 3+1 dimensions. Besides, if we're allowing the number of dimensions to vary, we're presumably also allowing the laws of physics to vary.
Concrete example: suppose it's proven that in GR, with 1 time dimension and >3 spatial dimensions, the orbit of a planet about its Sun cannot remain stable. That still isn't fatal because $f(R)$ gravity can conceivably replace GR in those universes and one of the theories might have stable orbits.
In other words, for the arguments in the quote to be convincing against extra spatial dimensions, they need to show that there is no possible theory in >3 spatial dimensions where the orbit of a planet around a Sun is stable - something which is presumably very hard if not impossible.
I'm wondering if I'm missing something.