Here's what I think:
It's like most of these miracle 'cure all' fads that appear. Absolutely no sound scientific evidence of success. It's just smelly oil. Other than that it's all a load of hype and marketing. The same as Tea tree oil, Aloe vera, Ginkgo, Ginseng, royal jelly, bee pollen propolis etc The one that gets me is Manuka honey, it's got a foul taste and they get big surpluses in New Zealand and Australia, so they market it as a health product; that can cure all ills. It's the only way they can shift it. Just like Neem oil the internet gets bombarded with bogus claims of it curing almost everything.
Back to the question A smelly oil will probably clear bugs but so will one hundred and one other things. Personally I don't put anything on my veggies. The only thing I find I can't grow is Rocket/Roquet, it gets eaten completely. Everything else generally does okay.