While the scene focuses on his guilt and regret, it also uses Lily's death to further develop his character.
In fact, some argue that he's actually not a very responsible teacher or Headmaster at all.
She, too, wasn't always on his side, which may be why the two ultimately fell out, and why Snape feels so guilty.
For the first six installments of the series, it's hard to believe that Snape is trying to protect Harry because he loved his mother, but when Harry learns the truth, a lot of interesting revelations come to light.
They recently made a comeback with refreshed pandemic-related material in 2020.
Snape was in love with Lily and can't move on because of his guilt.