Two objections, at the very least:

1. Running (many) chains in parallel reflects on the distribution of the starting values as we cannot be sure to "reach" stationarity for all of them in a finite number of steps. Hence a bias.

2. Weighting MCMC values by their likelihood means the likelihood is counted twice (as a power of two!), since the values are approximately distributed from the posterior, i.e., the prior x the likelihood. Hence another bias.

Now importance sampling may be associated with MCMC, as we recently [demonstrated][1].


  [1]: https://arxiv.org/abs/2207.08271#