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All this manual calculation rather than just invoking the isoperimetric inequality? I mean, manual calculation isn't bad, if you want to do it also as an additional demonstration, but I think the isoperimetric inequality is worth a mention here!

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The isoperimetric inequality is correct but too short. You need a longer proof with more equations and integrals to convince people.

The calculation is somewhat bogus since comparing a sphere and a torus with the same radius isn't really valid.

The author should either compare the surface areas of spheres and toroi with the same volume, or vice versa.

The author does indeed do that via their definition of the volume equivalent sphere radius r_s.

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