One need not discard any view, one need simply acknowledge that alternatives might exist. There is quite a bit of evidence available in journals such as the Journal of Parapsychology for example that strongly suggest alternatives. Further, many theoretical physicists are beginning to ask the most basic questions based on their emerging acceptance of the reality of anomalous phenomena, including most interestingly, macro PK.
The point is that one should not become enamored of any belief system, including that of materialism or scientism. I am not saying that scientific protocols should be abandoned, far from it. But I do caution that one must be sure that the problem at hand is consistent with the measurement capabilities available. When the data is not highly measurable and quantifaible, ie when it turns from quanta to qualia, existing techniques and assumptions prevalent in materialistic science will be less useful. This is more a statement of fact regarding the nature of physical science than a suggestion that it is useless. Some anomalous phenomena, such as PK, is quantifiable and indeed has been studied carefully in research laboratory conditions. The phenomena which is less amenable to such physical analysis involves studies of mind, belief and consciousness. Yet this is an expanding area of interest to many leading physicists, many of whom are charter members of the Society for Scientific Exploration (SSE).
As to the approach, we look for patterns in the data. We look for trends which acheive significance in various ways. We note phenomena that seem to share common ground and ask the obvious questions of 'why' and 'how'? We do not accept every report as valid; in fact various meta-studies conducted over the past several decades suggest that up to 90 percent of all such reports should be seriously questioned. So the notion that everything cannot be accepted at face value is supported. However, and this where skeptics get it wrong, this does not mean that everything can or should be discarded. Indeed what is studied is that other 10 percent or so, and in that we find some of the most interesting data which leads us to wonder how complete our basic theories truly are.
It is clear from the history of science that science is emergent. Physical laws are always in flux and likely will be for the forseeable future. Anomalies should be considered indicators that the physical laws need change. Heloicentrism was at one time an anomaly as was quantum mechanics. So holding that physical laws must dictate what is correct and what is not seems a bit short-sighted. Yet this is the very heart of many critics' position. An illumnating alternative is found in the scientific theory of Conscious Realism and Observer Mechanics. This is reading that any serious critic should consider carefully. You can see an introduction at: http://www.cogsci.uc...ousRealism2.pdf
Non-material events can be experienced. In fact that is the precise conumdrum facing science. It is incorrect to say that becuase something is unmeasurable it is de facto impossible. We cannot reliably detect gravitons, yet most theoretical physicists have no problem with their existence. We cannot measure hyperdimensions, yet mainstream physicists rely upon them to explain quantum mechanics.
Regarding 'how it works': one notion is that we are in fact interacting with content-rich information fields and it is our reactions to these fields which inform anomalous psi experiences. The other notion that is far more popular in the 'paranormal' community is that discarnate entities are responsible. I am not personally persuaded of this latter position as it provides no uniform explanation for most phenomena and abilities. It would seem more reasonable that a single basis should exist, and this is the strength of the information field interpretation.
As stated earlier, the key object to research is adopting techniques to reflect the realities of the data; understanding that not everything is measurable in the traditional sense. For example, in mediumhip research, the goal is to determine if anomalous information transfer occurs. This is done by carefully isolating the participants, including experimenters, and then seeking patterns in the data suggestive of such information transfer phenomena. Clearly we can use statistical analysis for this purpose, so in this sense we have measurement. But to date there is no measuring device for confirming the flow of such information although there are those in the neurophysics community quite interested in that problem. More importantly this example illustrates what is and is not scientifically demonstrable: we can (and have repeatedly) demonstrated anomalous information reception. What has not been proven (and frankly I have no idea how one could) is that this information must be coming from the deceased as many believe. So there are limits as you have suggested.
Science and spirituality can indeed be combined. Many people working in research Parapsychology are otherwise hardened scientists. But as a group have been mightily affected by the research and its implications. It is this realization which actually motivates many of us. And yes, there is a common belief that science, in some expanded form from what it is today, can and should help us understand more about ourselves, our universe and our potentials.