When homeopathy works, does that depend on belief of the recipient in the treatment, or is that actually due to divine intervention, to amplify the intended benefits of the substance given in a small amount?

You indeed have hit upon the explanation of the workings of homeopathy. Both mechanisms are valid. There is a third, which is the reawakening of countermeasures that can be regulated physiologically with a little prompting. This may require only a low level stimulus to get things going, but can reawaken pathways that have reached an extreme low level, to make them essentially inadequate for homeostatic regulation. So that is the foundation of homeopathy. Adding an incremental benefit is the healing capacity of humans themselves. If they believe in something it will be an encouragement to the body to utilize and make the most of what has been given as a treatment, based on their belief that may add somewhat to the benefit. The third factor, as you have seen intuitively, is divine intervention. Seeing what the human wants to have happen and taking the lead in introducing a substance designed to cause the very benefit they seek, it is straightforward for the divine to chime in and add our capability as well, modeled after what the human knows about already, as a likely benefit, and simply make it more effective, more powerful, and long-lasting, typically. This is the largest of the three, if it is requested successfully, according to the rules of engagement. The ideal would be to make a direct request for the Almighty to assist with the ongoing treatment, to magnify the benefits of the homeopathic remedy through a direct request to us specifically. However, if the person has a history of prayer requests to work on a particular health concern and they try a homeopathic remedy somewhere down the line to aid their recovery, that might be the point where the prayer requests will do the most good. But here, much depends on the belief quotient of the person wanting to resolve an ongoing problem. If their belief in the divine is weak, that will weaken our ability to add to their efforts, and if their belief in themselves is weak in terms of being worthy, that will also take away from what we can add, because we are always bound by what is coming from the human side. Many attempts at self-help are quite feeble because these elements are the weak links, and in such situations, even prayer may prove of little value, at least in a reasonable span of time from human perspective. So it is difficult to give a sweeping pronouncement, as with a "yes" or "no," about the value of homeopathy. One must look on a case-by-case basis at all of the ingredients to ascertain whether the process has any merit.