Palliative care, which is what we are talking about now, is different than treating a disease. Standard medicine is well aware that tending to the emotional needs of the patients, including providing an on-call chaplain at some hospitals, helps make some patients feel better. Hospital chaplains have been around for a long time, and are not specific to alt.med, nor did the practice of providing access to religious support derive from alt.med.
That said, it's interesting that the paragraph you quoted doesn't mention the word "prayer" at all. It refers to "spiritual well-being." Earlier in that page it pointed out that spirituality "is often expressed through religion and/or prayer"", but it doesn't say that it's the prayer which leads to any change.
More specifically, it also says that "Meditation, twelve-step work (as practiced in Alcoholics Anonymous and similar groups), and seeking meaning in life all involve spirituality. Even simple practices such as silent observation, listening, or gratitude can become part of an open-ended spirituality that can infuse everyday life. Some people express their spirituality by spending time with nature, doing creative work, or serving others."
It notably doesn't say (as you suggest) that prayer is an effective method to reduce stress, nor does it say that prayer is any more effective than 'spending time with nature' or 'doing creative work.' (To that list I'll add 'talking with a counselor.')
So I don't understand your point. That page doesn't show that prayer "works", except perhaps as pallative treatment for those who already believe that prayer is effective. But then that's not very surprising, is it? And other forms of palliative care would also be effective. Perhaps even more effective?
You must certainly acknowledge that "spiritual well-being" as given on that page is a rather broad term. Telling a patient to spend time relaxing in the mountains, as an example of spending time with nature, is not specific to alt.med. Indeed, a century ago, going to a spa in the countryside, where one could get away from the stresses of life was the common treatment for well-off people with tuberculosis, 'consumption', and other hard-to-cure diseases. So it's not like these suggestions are foreign or even recent to standard medicine.
(Though once we figured out how to cure tuberculosis, this treatment for tuberculosis disappeared. That's why palliative care is different than treatment.)
Also, don't you find the circular reasoning rather odd? As you quoted, "those who reported spiritual well-being were able to cope more effectively with terminal illnesses and find meaning in their experience", but as I quoted in the second paragraph, "seeking meaning in life [involves] spirituality." In a rather direct sense it says that those who seek meaning are more able to find meaning in their experience. Should anyone be surprised about that?
I also think the idea that one should "find meaning in their experience" is a biased term. What meaning is there in going through cancer treatment? Other than the obvious - a combination of genetic and environmental factors, plus random chance lead to having cancer, and it's pain in the ass to get cancer treatments - why should anyone seek a deeper meaning?
Of course, if you think the universe is constructed around you, perhaps because there's a God or universal spirit who intercedes on your behalf and 'works in mysterious ways', then perhaps thought, prayer, and reflection might help revel that hidden meaning, or perhaps help you accept that it's all for the best. But then it's rather biased metric, because those who don't think there is meaning to cancer will also tend be those who don't have a religious or spiritual faith.