If you think of a person's spirituality as the manner and degree to which they cope, deal with, or perhaps transcend their psychological self*, then one of the first things you might notice is that such a view implies everyone has a spirituality of one sort or another. There is no such thing as a non-spiritual person, in that view. Instead, there are just people with varying ways and degrees to which they deal, etc with their psychological selves.
I'm processing this perspective here. I don't disagree that spiritual development does in fact touch on what you are saying here inasmuch as one of the key factors of it is becoming "the Witness", where you are able to see your small "self", the psychological self as an object, rather than being stuck within it as the subject itself. Robert Kegan has a saying that captures this regarding developmental stages of growth that, "The subject of one level becomes the object of the subject of the next level or stage." The more we can "see" ourselves from outside of being that subject itself, the better we are able to handle that psychological self, recognizing it as more a feature of who we are rather than being the very definition of who we are, which is where most people as you say live.
I think there is more to spirituality than this however as such a state has the effect of actually transforming the psychological self. And also what is described here as "transcending the psychological self" can be done through normal development, or things like psychotherapy where you turn the subject self into an object self. Meditation is quite potent in this regards. The effect of this "transcendence" can in fact be experienced and described as a spiritual experience because of the liberation or freedom from the shackles of what held us captive. I would suggest spirituality more pointedly is that sense of Freedom one experiences from the lower self. And that spirituality itself develops in stages as well, as a separate line of development.
So in essence, this does agree with what you're saying here. Now to your actual question...
The question might then be put, "How effective are the various major and minor religions at teaching people to deal well with their psychological selves?"
This is a question I think about a great deal, hence why I'm spending some time in response. I am very much a believer, as it appears you are as well, that one's spirituality and one's psychological self are integrally tied together. I'm fond of the term psychospiritual to describe that relationship. When it comes to religions I think those who delve into the psychological fare much more strongly in that regard than those who deal more with simply the structures of the religion, its lineages, its teachings, etc. There has to be some actual practices than engage the person on that level, such as meditation or any of the basic "yogas". One can have a wonderful philosophy, but if there is no yoga, it doesn't really go anywhere. So a practice is key to this.
In the West such practices, such as Contemplative practices within the Christian church, fell to the wayside largely and replaced by a stronger emphasis on doctrinal issues, that being 'right' scripturally is of paramount importance. This has the effect of taking one out of the interior spaces of one's psychospiritual development into a much more cognitive space in which truth and reality is external to oneself. I think that has a lot to do with the rise of modernity itself with its emphasis on seeking out knowledge of the world through the sciences and analytical means.
As a result of this void of interior awarenes left in the wake of such a shift, the Eastern religions hold a great deal of appeal to those looking for that inner truth and reality which is 'felt' or sensed within them. I agree with you that all have this within them. But the tools aren't taught very well, if at all in the West. I think it's more than just a religion issue, but a cultural problem, a pathological underside of modernity itself.
The Eastern religions have much more as a part of the body of their teachings these psychospiritual tools, or technologies you can call them. And while they are there also within Christianity, you have to somewhat seek them out and in the worst cases rescue them from the religion itself. It's not something which is readily available for those who are on that path of psychospiritual development can fall into easily. It's more like you say, it's a matter of the student learning in spite of them having crappy teachers.
I wouldn't say these types of students, of which I include myself, are necessarily "gifted" in this regard. I think the impetus for them essentially teaching themselves comes as a result of some major life trauma or crisis which more or less breaks the door down for them and lets that Light shine through in such a way there is no other option available to them but to find it out for themselves at all cost.
I can talk of my own history here seeking out guidance into that spiritual space from those in the Christian religion who were clearly out of their depth and couldn't relate to the reality of which I was speaking of. Whereas in Eastern religions they would get it right away. That actually caused a great deal of delay in my own development as I looked for teachers who could teach but found none. If I were so "gifted", it wouldn't have taken me a few decades of figuring it out myself to come to where I am today.
The missing key for me the whole time was simply primarily one thing: Meditation. The other part of that too was more modern structures with which to hold and processes these things, understanding them in developmental spaces, rather than hanging them on the predominantly mythic-literal structures of Christianity. While Buddhism and Hinduism certainly have mythic structures, they do have and freely teach and emphasize the interior space development as well.
But for me as a very postmodernist and integral thinking person, the dominance of mythic structures don't work as well in supporting the spiritual awakening side of me with my cognitive structures. I need to hold these things in more rational and transrational structures, rather than predominantly mythic structures. Not that those are bad, but it doesn't fit within the modes of my thought structures. I don't think predominately within those frameworks, in other words. I need to translate things a little differently.
I'm afraid I may have strayed too deep into details here and missed your point, but as I said your question is a very good question, and like other questions you've asked recently touch on some very deep and complex areas.