Sure. But the square root and the sine function also have nice geometric interpretations.

Bring radicals don't. They're just defined as a solution to this particular quintic.

Kinda the similar story with the Lambert function.

The Bring radical has a great geometric interpretation: BR(a) is where the curve x^5 + x + a crosses the x axis.

Like sine or exp, it also has a nice series representation:

    sum(k = 0 to inf) binom(5k,k) (-1)^(k+1) a^(4k+1) / (4k+1)

    x <- x - (x^5 + x + a)/(5x^4 + 1)

Why not invite it to the table of functions?

Ellipses are simple and beautiful figures known to every child, but why do we rarely invite the elliptic integrals to the table too?

I guess my point is that "nice geometric interpretation" is a little subjective and hasn't led to much consistency in our choice of which functions are popular or obscure.