MCMC is good not just for computing probabilities but for integrating any function. Without it, scientists were limited to functions they could integrate analytically, or to well-behaved, low-dimensional integrals they could approximate as a series of trapezoids. With MCMC, they’re free to build complex models, knowing the computer will do the heavy lifting. Bayesians, for one, probably have MCMC to thank for the rising popularity of their methods more than anything else.
