I had the privilege of attending a public charter arts school for middle school and most of high school. There, among other activities, I studied piano from 7th grade to halfway through 11th (4.5 years). I've only played sparingly since then--usually just on a public piano to maintain the pieces I already know--but I've found that while my site reading ability has withered, my sense of timing and ability to improvize has matured substantially. I don't know many new compositions, but I can throw together improvizations of pieces I already know often much better than I ever could.
Having discovered the joys of improvization, I think [.85 credence] the majority of people who take up piano lessons should aim to become proficient at improvization first and put learning others' pieces second. Improvization--messing around with chord progressions and scales in a way that sounds nice--is easier to learn and provides easier satisfaction throughout one's life, even after lessons and formal practice has stopped. E.g. When you get off work, do you really want to struggle at the piano to learn a new song (or play one of the songs you already know and have played scores of times), or do you want to just chill and mess around in a more self-expressive way that usually sounds quite nice?
While learning piano 'classically' with scales and primarily only playing others' pieces will eventually enable you to play 'the Greats' (Clair de Lune, Maple Leaf Rag, etc.) well, doing so takes hundreds, if not thousands of hours, and is a much more laborious path. Unless you know you are the type who can sink 5+ hours a week for 2+ years, I think you should really consider starting the improv path first. The improv path is quicker to become proficient in, so you can always transfer over to learning classically later.