Exercise definitely does improve tons of factors, even if it is not an effective way to lose weight on its own. There is also, as I understand it, some degree of distinction between weight loss and weight gain prevention: while the saying "you can't outrun a bad diet" seems to hold true for people actively trying to lose weight (something I can personally understand, when I'm in a running training block my appetite is unreal), there's also a lot of evidence showing lower rates of obesity in more densely-populated areas, where people naturally walk more, and evidence of improvement when people move from low density to high density areas, which suggests there's a component of just moving around in daily life that can be a factor:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9378543/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6747269/ https://onlinelibrary.wiley.com/doi/10.1002/oby.23634 https://academic.oup.com/edrv/article/43/6/966/6532493?login...

Just do both and get the benefits of thinning out _and_ mental boosts from exercise.

Your body will burn fat in caloric deficit. No keto required.

Fit and fat is much better than unfit and unfat.