It's easy to design examples that trip MCMC, but they also trip straight Monte Carlo integration even more.
You can, of course, construct examples where MCMC will do worse than simple Monte Carlo integration, but these are uncommon. They mostly illustrate the difficulty of picking an appropriate jump size.
If you directly draw samples from the underlying distribution, you don't have to worry at all about whether the markov chain or other process (e.g. HMC) is fully sampling the distribution.
MC has no rejections and always samples the entire distribution. You simply don't need to worry about trajectories not going everywhere they should.
You can, of course, construct examples where MCMC will do worse than simple Monte Carlo integration, but these are uncommon. They mostly illustrate the difficulty of picking an appropriate jump size.