database schema for a product with multiple categories and hierarchical categories
What you are proposing is a good solution for your requirement of M:N products to categories and hierarchical categories.

To avoid exposing yourself to numerous updates:
You need to do two things to ensure that you don't have a lot of updates in your intersection table.  

First, you need to be sure that your categories have a stable, persistent primary key.  

Second, you need to link food items to leaf categories.  Don't join Cherry to Red, Healthy, Fruit and Food - just join it to Red and Healthy.  Your nested sets take care of all of the secondary (and higher level) associations.