PS-the non-drama special details to add: navigating their playlists, variety show options...
KBS's YT page= https://www.youtube.com/user/kbsworld
ALL the playlists=https://www.youtube.com/user/kbsworld/playlists
Kind of confusing, but WITHIN the playlists you'll see the KBS-CREATED PLAYLISTS=
https://www.youtube.com/user/kbsworld/playlists?view=1&sort=dd&shelf_id=0
That's where I got the drama specials and where various programs are that aren't full length dramas (those are licensed through kocowa etc)...
VARIETY SHOWS are in "KBS Entertainment" ie Hello Counselor (all of 2018 is available), "Somevival" Survival 1+1 in some countries, Season 4 of Happy Together in some countries etc... https://www.youtube.com/user/kbsworld/playlists?view=50&sort=dd&shelf_id=17
Lastly... "Entertainment Shows" don't have a lot that qualifies and some of the lists are out of order (and all seem to be newest-first, the opposite of how we tend to watch when making a playlist/marathoning, oh well?), BUT I would think Yu Huiyeol's Sketchbook and Immortal Songs certainly qualify for that one box. Keep in mind, a year's worth of Sketchbook or Immortal amounts to 65 hours of mostly live music with interviews between on Sketchbook, same length not sure the content breakdown on Immortal. (Sketchbook's by far my favorite music program FWIW-artists seem to be able to do their best in that place for some reason, and he brings out quirky tendencies of guests and gets them to be open and have fun, almost definitely because he's talented and listens very carefully/considerately to everything they play as a musician himself.)
There are 2 easy options for Yu's Sketchbook via this YT channel (I know KBS has another, but I am not subbed to the other): 2014 is complete, and 2015 starts in March BUT goes all the way to March 2016 (2016, '17, '18 are incomplete and 2019 aren't in a playlist though I've seen episodes lurking around if you're sticking to all legal ways of accessing which makes this a little harder, I'm guessing. Anyway, just an option; this isn't dictatorial (or wasn't-correct me if we'll get our hands ruler-smacked for being creative lol!) so I imagine ANY total year of content watched in a year, even if there are 12 missing from one and you grab 12 from another, is fine since it's crazy hard to keep up with as is! [I am gonna be a little more gentle on myself and find Lee Dong Wook Wants to Talk somewhere and probably also watch Jung Hae Il's Travel Diary there on KBS World if it counts!]
https://www.youtube.com/playlist?list=PLMf7VY8La5RFWmZwi2xcz8kJVkA_7e4bJ
There's also Immortal Songs 2- to get a year, you can go July 6, 2013 (earliest in the list)-July 5, 2014 or they have all of 2014, 2015, 2016&17 essentially (occasionally they remove one-you can find or replace it w/another to get a year's worth).
https://www.youtube.com/playlist?list=PLMf7VY8La5RGGFEmmcjKzJ4BCFdBuo4o8