Okay, first of all, incoming wall of text. I am going to start with a long overview that I feel is helpful in getting to answering your question. Skip to the second section if you want the more direct answer to Loop's post. However, some of it may be hard to understand without reading the overview. I learned most of this from my readings of Ken Wilber, who is regarded as one of the most influential American philosophers of our time.
--------------------------------------------Western nations developed under the control and influence of mainly organized Judeo-Christian beliefs. They controlled not only the way governments worked, but also the direction and guidance that philosophy, science, art, and cultures grew. At this point, the human race (focusing mostly on Western civilization because I don't know as much about Eastern during this time period) had yet to separate different areas of thinking. Instead of saying that astronomy and religion were separate, we wanted to keep them together and force them to coincide perfectly. This led to issues like the Church calling out Galileo as a heretic for speaking out against the church's teachings. This obviously created problems.
Then (well during for the Galileo example) we had the Renaissance! The invention of the printing press allowed people to learn and spread ideas outside of their Churches. Those previously mentioned disciplines burst out into the open and away from the control and guidance of religion. Art flourished, the study of ethics and the idea of personal liberty exploded into being seen as inherently just, and all realms of study other than just religion were accepted as mainstream ways of making progress with the human race. This was a incredibly large transcendence, and that's why it happened over 3-4 centuries. These types of immense evolutions of human consciousness cannot happen overnight. At the same time, each new level of transcendence also brings about new challenges. To save time, inserting Wikipedia link about issues that arose during this time:
"On the other hand, many historians now point out that most of the negative social factors popularly associated with the medieval period – poverty, warfare, religious and political persecution, for example – seem to have worsened in this era which saw the rise of Machiavellian politics, the Wars of Religion, the corrupt Borgia Popes, and the intensified witch-hunts of the 16th century."
RenaissanceThis means that society and the human consciousness needed to (and naturally does)continue on its constant journey of evolving and working towards transcending the Renaissance period, incorporating what we learned in that period, and propelling ourselves into a new age. Enter, the modern rational/scientific world view. This worldview created almost unthinkable levels of scientific and technological advancement. Think about everything that has been invented and researched between say 1800-2000. It is rather insane. Society started to value what we call rational, scientific, objective, observable thinking. This is a very powerful way of thinking when focusing on the exterior observable world.
Like other periods, this evolution created its own problems too. As we decided to value the exterior observable world as king, we disparaged the interior. Our consciousness, thoughts, and ways of thinking were seen as not important because they could not be looked at using our standard scientific model. We can look at a brain in a scan and easily study and see how it reacts and works to different stimuli. However, if we want to truly understand what someone is thinking, we have to talk to them and interpret what they say. Exterior things are observable. Interior things require communication and interpretation. That makes them much harder to understand and study. "Good" or correct answers to internal questions require not only the proper subject, but also the correct interpreter. There are better answers to non objective forms of research. For example, if you are analyzing what an art piece means, there are certain answers that are more correct.
This way of thinking is easily seen in our culture. We value engineering degrees a thousand times more than a humanities degree. I am not trying to say I disagree or don't understand why we do that. I am simply pointing out that it is true. So while the rational/scientific worldview has allowed for great progress, it has a flaw like all other periods of history. Its flaw is that we have chosen it at the complete expense of the interior.
I would say currently we are moving towards a view that is willing to incorporate more of the internal. It is clearly not widespread yet, but you can see it in certain areas. Businesses are seeing that creating work environments that lead to interior feelings of comfort and happiness increase productivity. Creating a community in your workplace, that has real non-hierarchical work induced relationships is seen as a necessity to recruit top talent (Look at Goggle, and almost any other rising tech company). So all we can do is see where it goes from here.
Bottom line is, in my opinion from reading these books, is that both the external and internal are incredibly important. They each have their strengths and weaknesses that they bring to the table. If you want to get a true concept of the world, it is imperative to include both ways of thinking as you approach the tough questions.
--------------------------------------------Now, onto looking at the OP more directly. I feel like the reason a lot of people are nowadays hostile towards the idea of spirituality is specifically because it is an internal force. Today's generation has been taught that the external observable world is king. That is why we are hesitant to believe in " seeking the states of mind that lie at the core of many religions" as you stated in your post. Most people also have a hard time differentiating between religion (or what I would rather call organized corporate religion) and spirituality. I believe that you can definitely be spiritual without religion. And more importantly, that the prototypical organized Judeo-Christian religions may actually stop you from being what I would call spiritual.
As Pooler mentioned, there are plenty of others like Harris who speak about these very high levels of consciousness. It is more prevalent in Eastern studies, but it can be found in the West as well. At this point, it seems rather ignorant of us to assume that all of these people are simply crazy.
At our level, we can still observe some sense of "Self". When you are thinking, you are able to analyze your own thoughts and notice issues or problems. In order to do this, you are actually stepping back and looking AT your thoughts. As in your thoughts become the object that you are observing instead of the subject doing the observation. This is a very low level of what people like Harris or Wilber are talking about.
It may simply be that their level is not something that is conceivable until you can conceive of it. An example:
Let's go back in history to when everyone thought the world was flat. If you were in this time period, and someone tried to tell you otherwise, you would say they were crazy. That is because your worldview would not allow you to conceive of such a notion. Everything you had observed in the world had told you that the world was flat, therefore how could you possibly think it was round? However, as soon as you went on a boat, or studied the sky, and were able to see differently, your belief on the matter would
instantly change.
I believe the same principle applies with these high levels of consciousness, spirituality and self. Until we go through the same exercises that these philosophers go through and experience the same things that they do, we are unable to see what they see. Remember, spirituality is on the internal spectrum. That means that it cannot simply be observed like something external (aka looking at a brain in a MRI), it must be interacted with and interpreted. That is why it is much harder to fathom and much harder to understand.
Therefore, I think his claims certainly hold weight. More importantly, I think we
must investigate and be more open to this way of thinking if we want to move in the most beneficial direction possible for our world. Once again, I by no means think we should get rid of our current rational exterior way of looking at the world. Simply that we should
also incorporate more interior ideas/thinking/studies.
--------------------------------------------Yes, I know this is a wall of text. No, I am not making a tl;dr section. This is intellectual discussion, if you want to read it go for it, if not I don't care. I wrote this more as a way to force myself to write out my thoughts as a learning process. I hope I at least addressed your OP in some way Loop. Please respond if you wish and I will continue to as well.