What I don't get about homeopathy is that there's not even a plausible mechanism for it to work. So research into its effectiveness is like studying how well rocks repel tigers[1]. It seems necessary to show that a homeopathic "medicine" is substantially different than plain old water first.
Many people who are into homeopathy have no notion of "mechanism". They don't know to ask such questions, or perhaps they don't care to.
In the worst case scenario, they accept lack of mechanism because "there are things science doesn't know", especially if their personal experience (random charlatan demonstration) is duped into belief.
This gets obfuscated by claiming that water has a memory and will "remember" those substances. Which is of course entirely implausible for anyone with sufficient physics or chemistry background.
You're right that no further study is needed for homeopathy, it simply cannot work at these kinds of dilutions.
The plausible mechanism is placebo which is mediated by the autonomous nervous system. The same mechanism at play with back pain via either chiropractic (placebo) and the majority of spinal surgeries (placebo).
Engineers and logical types take everything too much at face value (e.g. astrology and personality types, as well as health and fitness). The reality is usually that neither camp /really/ knows what they're doing, but have plausible (to them) explanations. Though empiricism is much more mentally rigorous than woo-woo/sham.
Generally, if the treatment achieves the patient's desired goals -- I don't think it matters (barring severe ethical concerns like body integrity dysphoria). People not generally educated in a Western background are more likely to be susceptible to placebo or "magic" (this includes the less educated in the anglosphere but also the majority in foreign countries).
Barring life-saving procedures, most elective medicine is -- imco -- on a similar, if not slightly higher standing compared to homeopathy et al.
Like any cult pseudoscience it relies on the placebo and nocebo effects.
It also does affect people for sure... For things like a very real ear infection... the a-holes will leave kids in agony for weeks before parents see a proper physician for antibiotics.
Note Psychoneuroimmunology is also interesting, but a more legitimate area of scientific study. It takes a deep scientific look into induced immune system disorders, and unlike cult nonsense it has quantifiable scientific studies (usually done with rodents) that can be replicated in your own institutional labs.
Cult Homeopathy is just expensive child neglect in my opinion... offering the worst outcomes for the naive and superstitious.
[1] https://m.youtube.com/watch?v=4GzMizVAl-0&pp=ygUTc2ltcHNvbnM...