Yes, you can fit this model using maximum likelihood or as a maximum a-posteriori version and get some estimates. Most regression software would also directly provide you estimates for $\pi_1-\pi_2$ with confidence (or credible) intervals based on the delta method. In contrast to drawing samples from the posterior, you are assuming that a normal approximation to the sampling distribution of the estimates makes sense (possibly after some suitable transformation e.g. to the logit scale). This is not the case when there are very few events. On the other hand, this should be fine, if $x_1$, $x_2$ and $x_3$ are all large, in which case the answers will probably not differ between approaches unless you use quite informative priors in the Bayesian approach.
However, note that you do not really need to do MCMC sampling in the particular case you describe, conjugate updating gives you an analytic posterior for the joint distribution of $\pi_1$, $\pi_2$ and $\pi_3$. You can then either draw independent samples from this distribution and form the difference $\pi_1-\pi_2$ in the samples (no worries about convergence of the MCMC chains). It might even be possible to analytically derive the distribution of the difference $\pi_1-\pi_2$, but I assume that would get rather messy, but numerical integration should be entirely possible.