While MCMC is certainly a useful method and has the potential to adress several difficulties that may exist with more traditional methods of parameter estimation, in my opinion it is not a magic bullet for several reasons:
1. The estimation procedure can be slow, especially for larger datasets.
2. The use of prior distributions may deter some researchers.
3. Sometimes it is not clear which prior distribution is appropriate.
4. Model specification in code requires a thorough understanding of the model in question from a formal point of view.
On the other hand, the potential advantages are:
1. Unified handling of a wide scope of models within the framework of graphical models.
2. Unified method for checking a model (posterior predictive checks).
3. Possibility of obtaining posterior distributions of derived statistics, such as effect size indices.
4. Model based handling of missing values and possibility of obtaining posterior distributions of missings in the dependent variable.
5. Possibility of rapid model prototyping.
For building confidence it may be useful to apply the MCMC method to the wide range of models used in the social and behavioral sciences, to cross check against the traditional criteria of parameter estimation (bias, consistency, efficiency) by simulation studies and to examine the influence of the prior distributions on the posterior distributions. But I guess this has already been done. I should check the literature.