Suppose we have a bunch of marginal distributions
which have some unknown joint distribution.
My understanding of the importance of copulas is twofold:
By always using the Probability Integral Transform to map our marginals to a multivariate distribution on $[0,1]^n$ with uniform marginals. It means that we need only spend our time compiling a list of joint distributions with uniform marginals, rather than the seemingly insurmountable task of cataloging many different lists of joint distributions, one list for each possible combination of marginals.
Copulas allow us to decouple the computation of the marginals from the computation of their dependence structure. This is important in higher dimensions when there is enough data to approximate the marginals non-parametrically, but insufficient data to approximate the joint distribution non-parametrically. Thus we instead choose an appropriate parametric copula to go with our non-parametrically estimated marginals.
We then simply take the appropriate copula and form
in order to recover a joint distribution with the appropriate marginals.
Now my question is, what exactly makes the Gaussian copula so important among all the possible choices of copulas?
Suppose our marginals are normal, and suppose that their joint distribution is not a multivariate normal. Then my understanding is that since marginals are decoupled from the copula, that their joint distribution (being non-normal) cannot have a Gaussian copula. But will they still be in some sense 'well approximated' by a Gaussian copula?
However I'm not entirely sure of the previous paragraph, as there is a family of distributions which are referred to as meta-gaussian, which I don't fully understand.
Furthermore, what if our marginals are not normal, now it seems even less justifiable to use a Gaussian copula, rather than just choosing the copula that fits the data the best in $[0,1]^n$ space.
Thus I'm left to conclude that the preeminence of the Gaussian copula is really about tractability, rather than stemming from any theoretical justification.