Here's a meta analysis[1] from 2012 showing that walking has a strong positive effect on depression. Here's one showing that biking and walking for commuting were associated with improved physical/mental health[2]. Here's a newer study showing benefits to the elderly from walking in urban green spaces[3].
These studies just shows that exercise improves mental health. I doubt anyone disputes that. The question is — are there benefits from walking directly, benefits that you can't get by say going to the gym. The claim of urban density advocates is that just being around thousands of strangers improves mental health.
Walking is basically free, and if you live in a walkable place trivial to build into a daily routine. Even if you exercise regularly but are otherwise sedentary you'll have improved outcomes from adding some walking. These seem to me to be obvious claims.
> just being around thousands of strangers
The nit I would pick here is that you aren't around thousands of strangers per se, but a large number of acquaintances in addition to those strangers. I see neighbors, local business owners, and the folks that work at shops and cafes nearby every time I leave my house in the city. I'm on a first name basis with many of them and we greet each other on the street. If I drove places I would miss out on the exercise and those small social interactions.
Seems like a well supported idea.
[1] https://www.sciencedirect.com/science/article/abs/pii/S17552...
[2] https://www.sciencedirect.com/science/article/abs/pii/S01604...
[3] https://www.frontiersin.org/articles/10.3389/fpubh.2020.5759...