The science is definitely mixed[1]. The proposed mechanism of action is logically sound, though, and probably worth exploring more than it is

[1]https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5845772/

It's possible the mechanism is plausible, but then how do we explain the fact that in most developed countries vaccination rates have gone up, yet allergy rates have gone up only in a few? Surely there must be some other factor causing it.

How do we explain the fact that countries with the highest growth of ice cream sales, have the highest growth of sunburn cases as well?

Hard to take a huge macro view of only two factors as you are doing and draw any meaningful conclusion. Hence the need for deeper study

Probably not all countries use peanut oil as an adjuvant.