First of all, there are no rules in "time travel" fiction. You really can not say that "they forgot how time travel works" or that they should follow the theory on "butterfly effect".
Secondly, characters realising "their past has been changed" "like adding a new movie to a library" where "you just see what changed" is not uncommon. You haven't watched a lot of time travel fiction if that was new to you or to borrow your own words, you "forgot how time travel works".
There are a hundred theories on time travel and how to solve the grandfather paradox. Time travel stories where it was simply a "butterfly effect" are the simplest type of time travel stories. The more complex ones use different theories and logic together.
For example, why do the characters suddenly receive memories when the past was changed? There are multiple time travel theories related to solving the grandfather paradox that they used here. One such is that a "change" in the past does not immediately take effect, there is a delay the farther you are from the source. Remember the gap is 20 years. Ripple effect.
As for how they are seeing the memories happen "live", one can argue that they took what is called the "literary freedom", meaning it was simply a medium of storytelling that they used to give an impact. In creative writing, the rule is always "show don't tell". It's better to show how the changes happening the past is affecting the 2013-present than to create scenes where the character is having a monologue explaining about how it affected them. Again, "show dont' tell".
However, I don't like using "literary freedom" as a reason why something is this and that. If I may, I submit that they are witnessing the actual changes happening in 1993-present because of another theory in time travel--when two time periods were connected because a time traveler changed something, the two time periods become both "present". This would also explain why the 1993-past suddenly became independent. Remember how the ML said his young self is doing things he can not remember? The reason is, the past is no longer the past, it is already a "present" too, independent.
And that is a problem the ML realised, since the 2013 is still a "slave to" the 1993 and those in 1993 are changing things on their own (meaning it's "present" too, not "past"), whatever they do on their own will affect them in 2013. You don't want that because suddenly you don't know what's going to happen, the past stopped being the past.
That happened after the ML left the two incense sticks in December 31, 1992. After that, the "past" switched to being the "present". Suddenly the characters in the "past" are changing things on their own, independently--without any influence from any time traveler. Those independent changes then reflects later in the 2013-present, and those in the 2013-present can not do anything about it.
It was why he called his younger self in episode 20 to forget that he exist. To not look for him. To not communicate with him. The two time periods--1993 and 2013--needs to realign or disconnect permanently. The ML even said that whatever his younger self decides will "make him".
It also explains why everything changed. In episode 20 they showed us for the first time the events between 2007 and 2012. This was to show things are already different. There are now two timelines, the previous timeline was wherein the ML died and did not return to his own present. The new timeline is the timeline from the new "1993-present". The final timeline we saw in episode 20 was the result of what happened in the "1993-present". When the old ML died, the "1993-present" got disconnected from the 2013-present where they can watch their new past memories happening live. The old ML said that he realized he is the incense. It means that he is the one that is connecting the 1993 and 2013 time periods.
It got disconnected when he made the final change in the past, tricking Min Young into making a promise to avoid him. While the old 2013-present Min Young still "saw" what happened, that timeline is already separate. A new timeline was created wherein none of them remembers the old timeline.
So what happened to old timeline? It probably still exists. There is a Min Young who was about to get married but the ML failed to return to his own time, and later this Min Young saw the last change in the past wherein his younger self made a promise to the dying ML.
In the new timeline, they lived happily ever after.
