This stuff is incredibly complex to study. And every time I read a study I forget to archive it somewhere. We don't have all of the answers, but studies like this one:https://pubmed.ncbi.nlm.nih.gov/37600508/
Really make me go, get your ass to the gym. No excuses.
And I suspect we are just looking at the tip of the iceberg here when it comes to things like BDNF. In short, exercise (specifically Strength Training and HIIT) seems well worth it since it is having some kind of effect on people who already have Alzheimer's.
I wish the fitness industry promoted the practical and real cognitive benefits of strength training instead of using using people with eating disorders and six-packs(six-packs usually aren't healthy by the way) to show/advertise the "perfect body". When in reality they should be promoting how much it can help your cognition and mood. I might get heat for this, but I'll say it anyway: I swear people would not have to take as many antidepressants in this country as they do if they just strength-trained twice a week. I see it in the software industry every day how people 15,20 years my senior (whom I respect since they are my mentors) and it hurts to see the beer gut and the lack of vitality (you can see this in someone's face regardless of age) just because as programmers we are sitting all the time, and if you don't fight the negative side effects of that with exercise your body will respond in nasty ways.
I understand weightlifting/resistance training is not for everybody. It is crucial to prevent sarcopenia a little too early in life. And it doesn't take a whole lot to get 90% of the benefits from strength training. Two sessions/week of 1hr each is all you need to reap most of the benefits.