$$1728 V^5 + F^3 = E^2 \;.$$

Can anyone point me to a concise, modern derivation and explanation of
the significance of the *icosahedron equation*, more modern and
concise than Klein's description in his book?

Lectures on the Ikosahedron and the solution of equations of the fifth degree. Felix Klein, 1888.

The equation first appears in Klein on p.62 (Dover edition) as $$T^2 = -H^3 + 1728 f^5 \;,$$ where $f$, $H$, and $T$ are "forms." $H$ represents the "Hessian form," and $T$ the "functional determinant."

I cannot find the equation in Jerry Shurman's 1997 book,
*Geometry of the Quintic* (PDF download),
although I admit I have only scanned his book.

Incidentally, here is a (crude—Sorry!) plot of $x^3 + y^5 + z^2 = 0$,
equivalent to the above by scaling variables:

Perhaps this equation has been studied in its own right?