His argument about children and their lack of genuine creativity seems a little misplaced - they are disparaged as 'tinkerers' rather than creators, but I can think of no better word than 'tinkerer' to describe Leonardo, or indeed many other great artists/scientists.
Aside from all that, I can think of plenty of reasons why anti-intellectualism would exist:
- not all forms of expert knowledge are equally valid. It was only around 100 years ago that medical expertise progressed to the point where it was more likely to cure than kill you.
- the value of expert knowledge/intellectual capital can be hard to determine. For example, once you move past the hard sciences, to sociology and literary theory and beyond, how much is contributed to human knowledge and wellbeing? How much power/status should be accorded to people in these intellectual disciplines?
- societies are composed of counterbalancing forces, between elements which are conservative/progressive, peaceful/aggressive, practical/visionary etc. No more than every business can be a startup, how can all people in a society consider intellectual investigation the main objective, as opposed to more mundane aspects of maintaining and running the society?
- science seems to have been stranded on one side of an ideological divide, where its support/derision is now a badge of political identity.