Copula models didn't really work like that in the mortgage-backed securities world. Mortgages had been packaged into securities for decades before the copulas became popular, and securities with exposure to prepay correlation (e.g. CMO sequentials) had been traded since at least the early 80s. For years, the view had been "mortgage default correlation is hard to model, and is probably pretty low anyway, so we won't worry about it." A fateful view indeed.

It wasn't until the creation of the ABX index and then TABX (tranched ABX, an index designed to mirror subprime CDOs) that people started to worry about correlation in mortgages. In early 2007, TABX market prices implied vastly higher levels of correlation than most people expected. That was one of the early warning signs to investors that their subprime CDOs weren't going to hold up in a stressed scenario. But almost as soon as those signals were detected, the market for correlation-dependent mortgage products disappeared.

Gaussian copula models were much more prominent in structured corporate credit. They became the de facto method of modeling default risk in portfolios of corporate bonds or credit default swaps. Gaussian copula models, and succeeding "base correlation" models were necessary to price all kinds of credit derivatives -- single-tranche CDOs, first-to-default baskets, whatever you want. Go to http://www.classiccmp.org/transputer/finengineer/ and take a look at the Merrill Lynch Credit Derivatives Handbook -- that's probably the best example of how banks used correlation models in the heyday of structured credit.

In practice, the Gaussian copula model became the credit derivatives market's analog of the Black-Scholes option pricing formula. The price of an option depends heavily on volatility; the Black-Scholes model provides a clean, mathematically tractable method of extracting the implied volatility from the price of an option. Once you obtain the implied volatility of several different options, you can decide which ones look rich or cheap, and you can use that implied volatility to determine a fair price for other similar instruments. The model has a number of empirical deficiencies, but practitioners were able to intuitively correct for them while trading. Gaussian copulas were the same way. Given the price of two tranches on a reference portfolio, you could estimate the implied default correlation of the assets in the portfolio. With that correlation in hand, you can price all sorts of similar assets. The problem is that copula models are significantly more complicated than Black-Scholes, and their deficiencies harder to correct for in practice -- the models had the effect of making traders rather complacent.

And finally, I hate the trend of bashing quants like David Li and blaming them for getting us into this mess. The few people that were making an intellectually serious effort to understand the mechanics behind the market are not the ones at fault. I think the effort to depict markets as ineffable black boxes beyond quantitative description is deeply anti-intellectual. The people most at fault are those who took the quants' research and applied it uncritically in the pursuit of short term profit, without understanding the math behind it.