How is it his responsibility to confess? Byul already made the boundary, Kang Lim is just respecting it. Its her responsibility, she made their relationship what it is now. She made Kang Lim be cautious. She is the only one who can reverse it.
I feel bad for her. Good job author. She's had a terrible childhood and it made her become terrible herself. But once you open yourself up to someone finally... And get "betrayed". It's a pretty shitty feeling