They actually do dance around a claim regarding cancer:
> Many essential oils have antioxidant properties. Antioxidants help prevent damage to cells caused by free radicals. This damage can lead to serious diseases such as cancer.
For comparison, Wikipedia:
> Aromatherapy may be useful to induce relaxation, but there is not sufficient evidence that essential oils can effectively treat any condition.
"Results of a 1994 studyTrusted Source found pure tea tree oil was equally as effective as the antifungal clotrimazole (Desenex) in treating fungal toenail infections. Clotrimazole is available both over the counter (OTC) and by prescription."
Reading through the wikipedia article it seems they're conflating all use of essential oils and specifically the use of essential oils in aromatherapy treatment, which is what the linked article is about.
"In laboratory and animal studies, the presence of increased levels of exogenous antioxidants has been shown to prevent the types of free radical damage that have been associated with cancer development" -- which seems close to what they were saying. A bit overstated for something published to the public, but technically accurate.
Overall it seems like the dismissal is a bias of the idea of essential oils, not necessarily a comprehensive review of the literature.
Honestly I'm WAY out of my lane here. I have no medical training and can't even call myself an interested amateur. So me trying to argue via random snippets of internet text would just waste everyone's time! :-)
The only thing I can really say is, there's a lot of medical-related material out there that seems vague and hopeful, that I don't think is doing much good for anyone but its creators. I'm with you - let's see what the research says and what the studies show, while keeping the placebo effect in mind. If crabgrass cures cancer, I'm down for some grazing.
> Many essential oils have antioxidant properties. Antioxidants help prevent damage to cells caused by free radicals. This damage can lead to serious diseases such as cancer.
For comparison, Wikipedia:
> Aromatherapy may be useful to induce relaxation, but there is not sufficient evidence that essential oils can effectively treat any condition.