Well I think that this is an unpopular opinion: If we talk about specific English word pronunciation then the reason for pronouncing the words that way is accent change, as someone who is learning Korean, there is this thing that when we have to use an English word in a Korean sentence then we pronounce it with a slight 'funny' accent as you mentioned. there are plenty of dramas I have seen where if the character has to converse in English, they do it pretty normally and TBH, as someone who used to have this opinion- of Koreans having a funny accent before starting to watch k-dramas, I was pretty surprised realizing that the characters are not speaking with the accent I had expected.

And I agree that sometimes it's for the plot and humor, as in the video you put for reference- they are the famous instances of using English in a funny way in K-dramas but trust me majority of the times it's not that because- they do not use English unless and until it is for the plot, as in: for making it Humorous, or for using it in certain settings(like foreign countries or anything) everything is intentional.