One of the biggest concerns with this perspective is that it can be used to argue that there is no truth and that everyone's beliefs are in fact equivalent. While beliefs and experiences are clearly real and personal, there is a long history in cognitive psychology of showing just how subjective they can be. Early humans thought it was obvious that heavier objects fell faster than lighter objects and that all creatures were created by some higher being (the history of creation myths and their adoption and modification by more modern religions is particularly fascinating). This is why we have science and math--these fields use the best methods possible to get at the "truth." I put scare quotes around that, because scientific theories, unlike mathematical proofs, are often only better and better approximations, not completely ground truth. Unlike math, our scientific theories continue to be refined, but instead of saying the old one was not true and the new one is, we instead look at their utility. Isaac Asimov, in his Relativity of Wrong essay eloquently explains how a flat earth theory is quite useful for many situations and that error over a few miles of flat vs. round earth is not all that great. Of course, if you want to go to the moon or put up satellites, the flat earth theory is not going to help much.

Interesting that you bring up Peirce, as he also coined the term "abduction", meaning "inference to the best explanation" in contrast to deduction and induction. Much of science and diagnostic reasoning is abductive. But abduction must depend on facts. This is where science and religion separates. As Feynman said, "Science is a culture of doubt. Religion is a culture of faith." Faith, by definition, is belief without evidence. Any abductive or inductive conclusion is only as good as the evidence. Taking evidence as true by faith is the religious approach, not the scientific approach. So to think that all beliefs and experiences are equally valid, might be very kind, but it is not very useful.

In fact, we are now at a dangerous time in the US and around the world where we have people (and in the US a president and large percentage of the population) making up "alternative facts" and denying other facts (such as the science of climate change). We have large groups pushing their religious views on others, even though there is no real evidence for the various creation myths and life-after-death promises of any religion. Telling people that all beliefs are somehow equal gives justification to the kinds of dangerous attitudes and behaviors that we now see. Senator Moynihan is widely quoted as saying "Everyone is entitled to their own opinions, but not their own facts." These days it seems that everyone feels they are also entitled to their own (often alternative) facts.