This paper revisits Gaussian Mixture Copula Model (GMCM), a more expressive alternative to the widely used Gaussian Mixture Model (GMM), with the goal to make its parameter estimation tractable. Both the Expectation Maximization and the direct Likelihood Maximization frameworks for GMCM have to grapple with a likelihood function that lacks a closed form. This has led to a few approximation schemes that alleviate the problem, nonetheless leaving the issue still unresolved. Additionally, past works have alluded to an additional challenge of parameter non-identifiability, but none has offered a rigorous treatment and a commensurate solution framework to overcome the same. This work offers solutions to each of these issues in an attempt to help GMCM realize its full potential. The source of non-identifiability is not only proven but also suitable priors are proposed that eliminate the problem. Additionally, an efficient numerical framework is proposed to evaluate the intractable likelihood function, while also providing its analytical derivatives. Finally, a view of GMCM as a series of bijective mappings from a base distribution is presented, which paves the way to synthesize GMCM using modern, probabilistic programming languages (PPLs). The main claims of this work are supported by empirical evidence gathered on synthetic and real-world datasets.