Nowadays, even the ABSRM syllabus is not entirely classical. For Grade 3, students have 48 pieces to choose from. I am learning Henry Mancini's Moon River, and a simple ragtime piece from the 1950s. And, I confess, the entirely classical Chanson Italienne by Tchaikovsky.
There is also a choice of Joni Mitchell (Both Sides Now), something from Oliver, a piece from Spirited Away, the Muppet Show theme, and Can You Feel the Love Tonight from the Lion King. Admittedly, even this is not really cutting-edge and modern, but they are recognisable tunes (well, not when I play them, but you get my drift).