while I'm all for MC ignoring their bio family if it was deserved, my understanding is that only ANY (dang that's an awk acronym for An Nanyi lol...) was involved in mistreating MC before. MC knew of AJ and AY but hasn't mentioned that he had issues with them before, mostly that he avoided due to association with ANY. why cut off people who haven't mistreated you without giving things a shot? I don't even remember a significant "ANY tricked bro x and y to abuse MC" situation, and in the current timeline it's already showed AJ still giving MC a chance after ANY said he was a bully and thief, and even then going against his ~precious younger bro~ when he locks MC out of their dorm overnight and makes him sick (also starting to suspect ANY in general).
Did bio family accept MC pretty fast? Yes, but considering bros 1 and 2 first considered him their new sister-in-law and got to know him from his dedication on the show, bro 1 at least was already shown to have an understanding of MC. Author also vigorously adds in ~unexplainable sense of intimacy~ with the bros and mom, AND when they get told he's their stolen kid ML is the one presenting evidence, including recordings and DNA tests that also showed that ANY was aware of it and repeatedly acted against MC. Additionally, AJ has gotten to know MC pretty well during the filming and said ML wouldn't be with someone who isn't good so while bumping up the relationship to brothers is a bit sudden he already had a good impression of him and could help things get going with the rest of the bio fam.

finally, the review is marked at chapter 89 which is literally the chapter everything comes out and when they meet up at the hospital ML had already secretly told the bio fam that while MC may say he's ok with ANY staying it's not appropriate or right given the events and they agreed, saying ANY had chilled them. When MC wakes up they cautiously meet and that's when the An fam ask if he's willing to consider them family, and then if he'd call them mom, dad, etc. so he calls them that from the hospital bed and it's meant to be a moving scene. Are they immediately happy family? No lol. MC is still insecure about it and still lives with ML while getting to know them so not 'barely knew them for a month then called them mom and dad affectionately'

ML fell in love with MC when they were children but MC doesn't know and ML doesn't know that MC wanted to find him/the man who was with him when he died after being reborn