Why don't we see Copula Models as much as Regression Models? Is there any reason that don't see Copula Models as much as we see Regression Models (e.g. https://en.wikipedia.org/wiki/Vine_copula, https://en.wikipedia.org/wiki/Copula_(probability_theory)) ?
I have spent the last few months casually reading about applications of Copulas. As I understand, Copulas allow you to create a joint probability distribution for several variables - and each of these variables need not have the same marginal class of probability distribution. For example : A Copula could be made to create a joint probability distribution of variables X1 and X2, where X1 is a Normal Distribution and X2 is an Exponential Distribution. Allegedly, this is quite useful for modelling complex and irregular real world phenomena that do not fully conform to "homogeneous and common" probability distributions.
In terms of applications, I have heard that Copula Models (i.e. the joint probability distribution produced by a Copula Model) can be used for a different tasks involving Causal Inference and Predictive Modelling. Since Copula Models are after all joint probability distributions, we can use MCMC Sampling to generate random samples from a relevant conditional probability distribution - and the mean and variance of these randomly generated samples from the desired conditional distribution can be thought of as the "predicted value" for a new observation (effectively performing the role of a regression model).
I have read the Copula Models are often used in the financial industry to model correlations and risk in financial markets, and instances where they are used in Survival Analysis for modelling dependencies in Survival Times - but apart from this, they do not seem to be nearly as widespread as standard regression models.
My Question: Does anyone know why this is?

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*My first guess as to why Copula Models are less widespread compared to Regression Models, is that the framework and mathematics required in Copulas is arguably far more complex compared to Regression Models. Thus, the potential benefits of Copula Models are never fully realized due to the complexity of the mathematics required in understanding them.


*My second guess as to why Copula Models are less widespread compared to Regression Models, is that far fewer software implementations exist for Copula Models compared to Regression Models. For example, I have seen some popular R packages that can be used for Copula Models (e.g. https://cran.r-project.org/web/packages/copula/copula.pdf , https://cran.r-project.org/web/packages/VineCopula/index.html , https://www.jstatsoft.org/article/view/v077i08 ) - yet these packages mainly seem to concern themselves with "fitting" the Copulas, and do not focus as much on how to use Copulas for prediction purposes (in the same context as one would use Regression Models). I came across an R package that allows for fitting Conditional Copulas (e.g. https://cran.r-project.org/web/packages/CDVineCopulaConditional/index.html), but it seems strange that this package requires you to fit a new Conditional Copula to the data according to your specifications - and does not allow you to generate random samples from an existing Copula.
Thus, are my assessments reasonable? Could these partly explain why Copula Models are not as widespread as traditional Regression Models?
Can someone please comment on this?

A short answer is that in practice for many applications we don't need the joint probability distributions. A cynic would say that it's also because the users don't event understand what is a joint probability distribution. A lot of applications of statistical modeling are in inference, such as medical studies, and they're interested in what causes certain outcomes. A regression is one of the tools used to do this. In forecasting applications in many cases users want to do scenario analysis, i.e. "what is y when inputs are x?" - these pre-specify x's and don't need to sample from their joint.
On the other hand, copulas are used a lot in some fields such as financial risk management (FRM) to obtain joint distribution of the factors. I'll show you one example that will help me answer your question.
In FRM you need to obtain the univariate probability distribution $F_y(y)$ of scalar losses $y$. Here's one way you could do it.

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*map losses $y$ to a vector of risk factors $\vec x$

*estimate a model $y=\mathcal L(\vec x)+\varepsilon$, perhaps, with a regression

*estimate the join distribution of factors $\hat F_{\vec x}(\vec x)$, perhaps, with copulas

*sample from $\hat F_{\vec x}(.)$ to obtain a set of vectors $\vec x_i$

*estimate the univariate probability distribution $\hat F_y(y)$ by fitting it to losses $\hat y_i=\hat{\mathcal L} (\vec x_i)$
Once you have $\hat F_y(.)$ you can obtain all risk metrics that you need.
You see how I used both regressions and copula here. So, as I mentioned earlier, in business forecasting our model users are interested only in $\hat y|\vec x$, i.e. "what is $y$ when inputs are $\vec x$?" In this case, as in inference applications, we don't need the joint distribution and copulas at all! We only need the [regression] model $\hat{\mathcal L}$, we can specify $x$.
FRM is one of the fields, where we can't specify $\vec x$ in many cases. We try to obtain their joint distribution $F_{\vec x}$. That's what copulas are useful for
