Not overly surprising as I haven't read the book - just the website. And the website presents a bunch of regression charts showing that, on average, the selected indicators are worse in selected countries with income inequality as defined by the authors - it shows no evidence that this affects 'everyone'.
You have missed MY point - yes, things may be worse ON AVERAGE for mental health, obesity, addiction rates etc. - but how does that extend to EVERYBODY? (e.g. how does it affect a well-off, slim, mentally sound, unaddicted, educated, physically healthy person from a family with similar genes who will most likely end up similarly?)
Social mobility works both ways - so, again, what is the benefit to the 'haves' of trying to change the social system of their country so that there's a greater chance their children will end up poor - and more prone to obesity, teenage pregnancy, lower education etc?
The more I read about the research, the more flawed and less convincing I find it.