You have to segment the claims in two:
Claim one (We have eliminated mechanisms aside from contamination for the obesity epidemic) rests essentially on anthropological studies with "uncontaminated" populations and conversely on the fact that animals in close contact with humans are also becoming obese rapidly even when holding calories constant (e.g. lab rats)
Claim two posits some substances that might be responsible, and I think roughly is oriented around:
a) antibiotics are intensively used in animal cultivation in a way that might leak into the food supply and water table downstream.
b) PFAS accumulate biologically after exposure at parts-per-billion levels, and become increasingly more concentrated as you go up the food chain and downstream, but e.g. potatoes are not exposed to enough PFAS to hit the thresholds for deleterious effects.
c) I don't think that lithium is actively taken up by plants to the degree that would affect pure unprocessed plant consumption, it's mostly about direct exposure and water contamination downstream.
I think the claim one hypothesis is really strong. I've had a similar hypothesis for a few years now, though mine was more oriented around infection than contagion, but similar in terms of structure - the more people you put close together the more likely they are to cross-contaminate with some substance that makes them obese, and the clearer difference we have between baseline humanity ~100y ago and modern ones suffering from the obesity epidemic.
I think the claim two hypothesis is a little bit weaker, but it's interesting that they have 3 proposed mechanisms that range from known effect, known presence (antibiotics), known presence but unknown effect (PFAS), and unknown presence but known effect (lithium), so I think they roughly descend in order of likelihood of explaining the obesity epidemic. I would guess, obviously, the answer is "all three", based just on the severity of the epidemic, but I'd guess it's quite likely antibiotics have a small effect, and also somewhat likely PFAS have a large effect, while I think the lithium hypothesis is the weakest of the three, but would be interesting to analyze further.