So the main argument in the article is the claim that your total energy expenditure is more or less constant, and independent of "exercise". In the sense that if you spend 500kcal on running, your body will "not do" something else it would have done otherwise (e.g. stress about stuff) that would have cost 500kcals.

In other words it challenges the idea that if you run today and not tomorrow, your TEE on day 1 will be 500kcal higher than day 2. The claim is that TEE on both days will be the same.

Ok, that is an interesting finding. But this sounds very specific to aerobic exercise.

The article then mentions how they monitored an active indigenous population, and were "surprised" to find out that "when controlled for non-fat body mass this active population had the same TEE as western sedentary populations" (emphasis mine).

Why is this at all surprising. We've known for ages that muscle increase leads to a basal metabolic rate increase.

The article tries hard to give a visual impression that two similarly rotund people (one from fat and one from muscle) would have an equal TEE, but if you dig into the claim, it's exactly the opposite. In order to get that non-fat body mass difference, you need to exercise. Its just that this needs to be anaerobic rather than aerobic.

We also know that obese people have higher BMRs than their thinner counterparts. Now if the claim in this article is true, then it would imply that the increase in BMR in obese people is exclusively due to a parallel increase in muscle. But we know that muscle increases with obesity. So again, not very surprising.

So, the claim "exercise doesnt work" indeed lacks nuance. Particularly in not addressing aerobic vs anaerobic exercise.