10-Day South Korea Itinerary Review (Seoul + Jeju) – Anything You'd Change?

Planning a 10-day Korea trip in October. Priorities are culture, scenery, shopping and food. Not interested in nightlife or hiking. Using only public transport/taxis.

Current itinerary:

Oct 8: Arrive Seoul → Hotel/rest → Gwangjang Market
Oct 9 (Hangul Day): DMZ Tour → Hongdae (if time) → Myeongdong
Oct 10: Gyeongbokgung → Bukchon → Insadong → Samcheongdong
Oct 11: Relaxed Seoul morning → Fly to Jeju (Stay near Hamdeok)
Oct 12: South/West Jeju (guided tour: Jusangjeolli, Jeongbang, Cheonjiyeon, O'Sulloc, Aewol)
Oct 13: East Jeju (Hamdeok, Woljeongri, Seongsan, Seopjikoji via Kakao Taxi)
Oct 14: Fly back to Seoul → Myeongdong shopping
Oct 15: Starfield Library + COEX → Final shopping & packing
Oct 16: Fly home

Questions:

• Does the pace look reasonable?
• Is Hamdeok the best base for Jeju?
• Bus tour vs private taxi for South Jeju?
• Is Hongdae after DMZ realistic?

Anything you'd replace without adding more travel?

Thanks! Looking for suggestions before I lock everything in.