This is something I never noticed...you are absolutely right in that there is no place to enter a promo code for a multi-city itinerary, even on the Flight Finder page.

The Flight Finder page, which has a place to enter the promo code, shows travel as either Round Trip, One Way, or Multiple Cities, but only allows Round Trip or One Way to be checked; clicking on Multiple Cities redirects one to the old booking page, with no place to enter the promo code.