First of all a "No placebos do not work" statement is made, promptly followed by admitting the placebo effect is real. Make up your mind, if the placebo effect is real then by definition placebos do work.
The issue here is what does it mean for a treatment to "work?"
If you take twelve people with flu and give them no treatment for a week, twelve of them will get better: reduced fever, reduced pain measures, improved nasal airflow. Does this mean "No Treatment" works? If you just tell them every day that they look much better and air out their room, many of them will report higher scores on comprehensive well-being surveys and lower perceived discomfort within days. We should be careful about our language, and I think that means to distinguish between "have an effect" and "work."
Many interventions have effects - positive of negative - that are reproducible and quantifiable. Placebos, aroma therapy, life coaching... That doesn't mean the work.
An intervention "works" if it produces the effect in claims by the mechanism it claims. Western medicine mechanisms involve interactions among molecules. For example, insulin lowers blood sugar because it stimulates glucose uptake by muscle. You can measure each of those molecules; you can measure their interactions; they induce phenomena consistent with the health outcome.
Homeopathic treatments claimed mechanism is "like cures like," and that water memory of exposure to a toxin allows it to displace the miasm causing the actual distress. Miasms aren't directly measurable and don't produce consistent effects. "Memory" in water or alcohol of past exposure to dilute toxins has no measurable effect on the chemical or quantum states of the molecule. There is no way to determine whether a homeopathic treatment has the effect it claims by the mechanism it claims. There is no objective way to prove that it "works," but its claims are inconsistent with otherwise proven science and chemistry.
So far as I know, no one has a mechanism for placebos. It's a reproducible phenomenon in an incredibly complex system, and science has generally bees satisfied to describe it as "the placebo effect." Placebos don't "work," but they do have an effect.