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 reason might be that regression and copulas do not answer the same question. Copulas are about the joint distribution while regression is about a conditional distribution or just the conditional mean, depending on how you look at it.
Yes, copulas are in a sense more general, as you can derive a regression function from them. But except for the most trivial cases, it would be a fairly involved exercise that would not give a closed-form answer. Then to be able to "see" anything or to get some intuition about the conditional distribution or the conditional mean function, you would need to simulate from the copula. And you do not always have the hardware and the software handy for that.
A regression, on the other hand, gives a very straightforward answer to the conditional mean question. It delivers an a solution that is much more easily understandable and much easier to visualize in your mind.*
So for the purpose of regression (conditional distribution, conditional mean), regression is just much easier to use. And for the purpose of copulas (joint distribution), regression cannot substitute for copulas. But apparently the interest in a joint distribution is not that common? (I end with a question mark, as I am not sure whether it is the interest that is limited or our apparatus that is inadequate / too complex.)
Regarding Ben's answer pointing to the historical reason as the most important one, I wonder if that is the case. Trying to imagine what would have happened had copulas and regression started simultaneously, I still see regression winning the popularity battle due to its relative simplicity as well as sufficiency for a task (modelling of the conditional distribution and/or the conditional mean) that is broadly relevant.
*I said more easily and easier which does not mean easy.
