Intense two week itinerary for next spring

Hello!

I'm planning an itinerary for a two week trip across South Korea next spring. I'd love your feedback on my current plan!

For context, my partner and I are active travellers and do not mind waking up at 6 or 7 am to make full use of the day. Our favourite activities while travelling are seeing UNESCO world heritage sites, visiting the local natural wonders, and hitting the speciality coffee scene ☕. We don't really enjoy markets/window shopping.

With that said, here's the current draft:

Day 1 --- Sat, 27th March --- Arrive in Seoul.

• Land around 7pm local time at Incheon.
• Reach the city centre via train/bus (~1 h). Check in.
• Walk around the local area, get dinner.

Seoul night 1.

Day 2 --- Sun 28 Mar --- Seoul palaces & UNESCO core.

• Gyeongbokgung, changing of the guard.
• Changdeokgung Palace (UNESCO)  + Secret Garden.
  • Secret Garden needs a separate timed ticket, book ahead!
• Jongmyo Shrine (UNESCO). 
  • Walkable from Changdeokgung, timed guided entry most days
• Bukchon Hanok Village.
  • End day here.

Seoul night 2.

Day 3 --- Mon 29 Mar --- DMZ day trip.

• Guided tour to the DMZ area: Third Infiltration Tunnel, Dora Observatory.

Seoul night 3.

Day 4 --- Tue 30 Mar --- Suwon Hwaseong + fly to Jeju.

• Morning day trip to Hwaseong Fortress in Suwon.
  • ~ 1 hour by public transport, then the wall circuit 2-3 hours.
• Flight Gimpo -> Jeju ~ 1.5h.

Jeju night 1.

Day 5 --- Wed 31 Mar --- East Jeju.

• Morning hike to Seongsan Ilchulbong.
• Manjanggul Lava Tube.
• Cycle part of the Olle trail.

Jeju night 2.

Day 6 --- Thu 1 Apr --- Hallasan + fly to Busan.

• Early morning start to summit Hallasan.
• Late afternoon flight to Busan.

Busan night 1.

Day 7 --- Fri 2 Apr --- Busan.

• Haedong Yonggungsa Temple
  • Start early to avoid the crowds.
• Gamcheon Culture Village.
• Beomeosa Temple.

Busan night 2.

Day 8 --- Sat, 3rd April --- Busan to Gyeongju.

Early morning: KTX Busan -> Gyongju ~1.5 h. Check into the hotel.

Afternoon and evening in the Gyeongju Historic Areas (UNESCO):

• Daereungwon royal tombs.
• Cheomseongdae observatory.
• Donggung Palace & Wolji Pond
  • Go after dark for the best views.
• Woljeonggyo Bridge, illuminated at night.

Gyeongju night 1.

Day 9 --- Sun, 4th April --- Bulguksa, Seokguram, Yangdong.

• Bulguksa Temple (UNESCO).
• Seokguram Grotto (UNESCO). 
  • Up the mountain from Bulguksa, bus or ~2.5 km walk.
• Yangdong Folk Village (UNESCO).

Gyeongju night 2.

Day 10 --- Mon, 5th April --- Gyeongju to Andong.

Train ~1h to Andong. Rent a car for local sightseeing.

• Hahoe Folk Village (UNESCO). 
  • Maybe catch the Mask Dance performance?
• Dosan Seowon (UNESCO).
• Take the last train/bus to Seowonju.

Seowonju night 1.

Day 11 --- Tue, 6th April --- Seowonju to Sokcho to Seoraksan.

Train from Seowonju to Gangneung. Rent a car.

• Abai Island and lunch in Sokcho.
• Stay in Osaek Hot Springs area.

Seoraksan night 1.

Day 12 --- Wed, 7th April --- Seoraksan.

• Seoraksan National Park hike.
• Hot spring in the evening.
• Drive back to Gangneung.

Gangneung night 1.

Day 13 --- Thu, 8th April --- Gangneung to Seoul.

• Morning train to Seoul.
• Remaining Seoul attractions:
  • Namsan
  • Insadong
  • Tea ceremony
  • Live music to end the day?

Seoul night 4.

Day 14 --- Fri, 9th April --- Departure.

• Morning flight back to London.

Is there anything obvious we missed? Is the plan way too ambitious (we averaged 25k steps per day on our last trip to Japan)? What would be your top museum recommendations for Seoul? What're your favourite coffee shops we should visit?

Many thanks!