It's just plain not true. Claims like this coming out in the past few years are based on Herman Pontzer's research in which he measured activity levels and energy expenditure among the Hadza, a forager group in Africa, testing the hypothesis that they'd be using a lot more energy because of how active they are. He found their energy expenditure on average wasn't actually much different than the average for sedentary Americans and developed something called the constrained energy expenditure model. This just postulates that the human body has various ways of compensating in the long term for high activity levels. If less energy is available for basic life processes that are not critical to immediate survival, those process will be modulated downward. Importantly, one of these is inflammation, which is likely why exercise reduces inflammation. At the very extreme end of this, you see things like Marshall Ulrich when he was attempting to break the record for time running across the continental United States seeing things like his hair and nails stopped growing. Female endurance athletes commonly stop having periods. Male endurance athletes often have reduced testosterone production.
But this compensation is nowhere near 100% and it also isn't clear to what extent this is mediated by food availability and possibly the intensity of the activity. The Hadza don't have a lot of food and they're spending all day walking around. Similar studies have been conducted on the Amish, who are doing far harder manual farm labor and have enormous amounts of food, and these studies found much high energy expenditure among the Amish compared to post-industrial sedentary Americans, as well as an average 9% bodyfat for males for whatever that is worth. Similarly, the experiences of endurance athletes with bad health symptoms like amenorrhea and low bone mineral density are limited to people who feel pressure to be as small as possible and don't eat anywhere near enough food. Those who simply eat more don't have the same experiences.
As you stated, we also have quite a bit of clear cut existence proofs that energy expenditure is not simply constant among all people regardless of activity. Pontzer himself has studied some of these extremes to figure out how much energy a human actually can expend. From what I recall, it seemed to be around 3.5 times whatever your base metabolic rate is, at least in the long run. Over short bursts, energy expenditure can be up to 10 times base. The greatest longer duration energy demands he has seen in the field is the Tour de France for men and pregnancy for women, both of which are about equal and seem to represent the limits of what humans can do. Obviously, the people doing these things are eating far more than they would be if they weren't doing those things. Nutrition advice for athletes is nearly the opposite of what it is for the obese and sedentary. Eat sugar like there's no tomorrow. Get as many liquid calories as possible. Avoid high fiber because it'll sit in your gut forever slowing down all other digestion and making you uncomfortable. There's a reason for this.
Two of my favorite podcasts, Stronger by Science and Iron Culture, was in the former case and is in the latter hosted by one of the researchers who works in Herman Pontzer's lab, and he is constantly expressing frustration over how the findings and model get misrepresented by the time they telephone down to pop science communicators and diet influencers. The model simply says that expending X calories per day in exercise will not result in a net difference of X calories expended in total. It'll be some percentage less than 100. But it won't be 0%. Exercise and activity don't make no difference at all.
I'd never heard of this Kurzgesagt thing, but it appears to be a group of animators that make cartoon explainer videos of basically everything? That might be entertaining but is probably not the best way to learn about the frontiers of contemporary nutrition and exercise science.