Huelsenbeck 2003 : Simulated Character Histories

When I read Huelsenbeck 2003, I was unsure why they were using the simulation-based approach to determining probable character histories. It seemed that running lots of simulated character histories and throwing out everything that did not have the observed character states on the terminal taxa would be computationally intensive. I knew there were ways of calculating the likelihood of ancestral character states, and I wondered why they were not doing that instead. Is it quantification of how frequently that state is present in consistant character histories is more readily interpretable than likelihood ratios in terms of how much confidence should be placed in that value? How does this compare to methods of generating confidence intervals?