Height cannot be Gaussian if only because height is never negative but every Gaussian density is positive for all real numbers, including negative real numbers.

"interactions of many genes that are measureable"

The interactions of many is commonly used to justify a Gaussian assumption, but there is no such theorem. The main theorem to get a Gaussian density is the central limit theorem which usually assumes an infinite sequence of independent identically distributed random variables (plus some), i.e., i.i.d. The i.i.d. is asking a LOT!!! Asking so much that in practice it is essentially unrealistic.

What people often mean by Gaussian is just a central peak, some symmetry, and long tails, but an actual Gaussian distribution has a lot more, e.g., sufficient statistics (as in a classic paper by Halmos and Savage, with a derivation in M. Loeve's two volumes, from Springer, Probability) which, as I recall, E. Dynkin showed are quite sensitive to deviations from actual Gaussian.

Gaussian is important in a lot of derivations -- a LOT is known. But in practice Gaussian has some utility but only as an approximation where don't need to be very careful about accuracy.