Yes, I would like to keep them separate, but I would not like to have a big prominent link to them at the top of my inbox. See the labels listed in the mobile app screenshot ("Family," "Fun," "School")? I (and most other people, I would think) would much sooner like to have big buttons to those things at the top of my inbox. Not "Promotions."

I think you under-estimate how much people care about these types of e-mails. At work we manage the mailing list for a big restaurant chain. Their "active users" subset of their mailing list makes up about 1/3 of the total, and that 1/3 opens 40% of the promotional e-mails they are sent within 2 days. For a lot of people "Promotions" e-mails both represents a leisure activity (hunting for bargains), and social activities (lunch with colleagues, dinner with friends, evening at the theatre etc.).