I think the scientific method is key; there's nothing wrong with it in principle. The trouble is, it isn't employed uniformly or even necessarily honestly. The basic idea is to examine the world and allow the information it gives us to be seriously considered. The issue is that at this point, the predilections of observers often cause them to make a judgement not to seriously consider it. We're all guilty of this, but part of scientific training that is under-emphasised is the capacity to be aware of one's own biases and not to allow them to cause rejection of data out of hand. One may not be able to avoid subjective bias, but one should, if one is to call oneself a scientist, or at least to claim a scientific approach, be able to recognise and acknowledge such bias and regardless, proceed with a truly open mind.
IMO, the main philosophical position that promotes this is agnosticism: the acknowledgement that we do not know anything with certainty, and indeed, that idea is deeply embedded in the philosophy of science. The day we feel certain about something is the day we stop investigating it: the issue is closed, the science is settled, and anyone who questions it is an idiot, despite the fact that the only way science progresses is when some "idiot" does in fact question it and demonstrates that it's a misunderstanding.
That's why I said in a recent post that science can never prove anything: it makes progress only by disproving prior misunderstandings, and everything is in some degree or manner misunderstood. On a good day, you'll get even the most biased person to accept this, but that's a very different matter to getting them to apply it in practice. It's not always because they are stupid or mendacious (though Ben Radford in Alex's clip appeared to be so), but because they haven't registered and acknowledged their own biases. Part of scientific training should be to develop scepticism in the truest sense of the word, with particular attention to being sceptical of oneself, ruthlessly identifying and taking steps to offset such bias.
IMO, the scientific method, were it always applied like this, would be applicable in all cases, including the evaluation of the psi/spiritual experiences of self and others. It's just that most people find it very difficult to live in a world where there's absolutely nothing certain to hang on to. At a certain point in trying to exercise agnosticism, one finds oneself in a state of recognition of ignorance, of how inexplicable the world really is, and can accept that without being perturbed. Einstein said the most incomprehensible thing about the world is that it is comprehensible. On the contrary, I'd say the most comprehensible thing about the world is that it is incomprehensible, and the most amazing thing, that despite our always mistaken understanding, it's nonetheless possible to evolve and learn.
I do question the reality of psi phenomena in the sense of knowing what they actually are. So, for example, I don't question that NDEs occur, and it's true I have ideas as to what they might represent, but I do question whether interpretations by myself and others of these/other experiences are correct. Sometimes when I read what people have to say about their experiences, I think they may be creating a narrative to explain them, and all narratives possess a degree of literalism that can effectively close off different avenues of enquiry. This is the counterpart of how some scientists try to squeeze the data into a preformed mould, as was illustrated by the one in the clip Alex played where the guy introduced the idea that below the cortical mantle, some brain activity was still occurring, as if that solved the problem of how the patient could see the colour of the suit a doctor was wearing.
I guess I'm trying to say that, without being stuck on stupid, there's still a lot of scope for genuine discussion, and there are marked differences of opinion amongst proponents. It's this area of discussion that most interests me, and from time to time, discussions here on Skeptiko have caused me to revise my opinion. Since coming here, for example, I've changed my leaning from panpsychism to idealism (thanks to Bernardo Kastrup's contributions), and also revised my views on the issue of alien abductions, which at one time I more or less completely dismissed. It's vital, in my view, that we can have such discussions without the intervention of those who are "stuck on stupid". The new Mod+ category is helpful here, but I would like to see the possibility of members being able to raise threads in that category: it's not clear to me whether/how that could be done.