It is quite a frustrating process to watch.
He has asked for help numerous times but without giving much detail on what he has tried. For example, he said he had tried "paleo". But paleo is a very vague term. People suggest things and he advises that he already tried them without giving details on exactly what he did.
He has not published blood tests or other diagnostics etc as far as I can tell, so it is very difficult to know what the problem which he calls "metabolic disprivilege" is. Do his close relatives have particular weight problems? What are the genetic tests showing in terms of genes known to inhibit fat mobilization? Does blood sugar go down when dieting? We do not know this. I would expect rationality would dictate a rapid exploration of this problem but I do not see it.
As many (like me) who have lost a lot of weight will have experienced, the scope for self deception and for the lower brain to hoodwink you is huge. People typically believe that while dieting they are less productive. I also believed this but by chance it turned out I have been collecting metrics on my productivity for another reason, and was able to find out that my productivity (on mentally hard things) went up substantially. But we don't know how he measured his productivity, if at all, and whether his claimed reduction while on the diet is real or not.
We also don't know whether his expectations about how much productive hard mental work you can actually do are realistic. I get the impression that he feels that if he cannot concentrate really hard on difficult things for 12+ hours per day, something is wrong. But very few, if any, people can do this. (not doing routine things, but e.g. learning new difficult things or trying to prove hard theorems etc).
I agree that some people find it easier than others to lose weight. And there are some rare people who have extreme difficulty or who are exceptionally prone to obesity. This has been mentioned as far back as Dr Atkins' original book. People who have more difficulty than most - like me* - in losing weight constitute about 20% of the community. The hard cases are quite rare. So some good quality evidence is needed to conclude that a given person is one such.
*My weight problems are IMHO partly a result of high (measured) endogenous levels of the hormone cortisol - 60% above the top of the normal range - possibly a result of some things that happened when I was very young, which increases appetite, increases insulin levels thereby inhibiting fat mobilization, and increases fat storage and makes it harder to maintain lean body mass on a diet. So I am not unsympathetic to those who struggle. Nonetheless, whatever challenges you have, a rational approach to a solution is best.