Alright, there’s a lot to unpack there. First, anything you may be interested in can be cast as an expectation. The posterior median, or 50% quantile, for instance, can be seen as the expectation of an indicator function. Again, the discussion about the typical set is about what it means to efficiently sample from a distribution. To obtain good samples of \pi is equivalent to having samples that belong to \mathcal{T}. That’s what it means to sample from a (high-dimensional) distribution. So, under normal circunstances, HMC does sample from the correct distribution and those samples are all you need to compute whatever quantities are well-defined (i.e. expectations of measurable functions which actually exist).You’re fixating on an illusory distinction between the (target) posterior \pi defined on a sample space \mathcal{X} and its typical set \mathcal{T} \subset \mathcal{X}. Maybe reading this by @Bob_Carpenter will help - don’t get thrown off by the catchy title and please heed his point (2).
Now, there are a few subtleties in all this. For instance, you can sample from a Cauchy distribution, but it doesn’t make much sense to compute the “posterior” mean, for instance. See this. In general, when things do not have well defined expectations, we can’t employ (a variant of) the central limit theorem to analyse MCMC output, and things get complicated really quickly. See this an links therein.