The following post may contain spoilers: Lilith becomes much more pleasant as the plot progresses (end of 0.11). At the beginning of 0.12 there is a small slip again, but generally she becomes well bearable.
I think that Lilli is not such a badly written character. Her problem is that she is someone with a lot of complexes:
She was a daddy's girl. When her dad took off in the middle of the night, it left her with a huge emotional hole. While she quickly became friends with Abby, Frank was always the intruder in her family because she still had hope that her father would come back one day. In Lilith's Longing, that's Lilli's side story, we learn at the end about some interactions from when Shannon and Frank were still married and he was very caring. So I think that Lilli didn't act that way towards Frank because she's a bitch, but simply because she had the naive hope that she would see her father again, and secondly maybe because she was afraid that if she opened up to him emotionally again she would get hurt if he ran away, which she probably doesn't think is unrealistic since she herself is aware of how she acted.
For the same reason, she is very protective where Abby is concerned: she knows what it's like to be emotionally hurt by her own father and does everything in her power to avoid Abby experiencing the same.
Other complexes: She is very aware that her body is not nearly as proportioned as Abby's or Becca's. While I personally don't see this as problematic, in Lilith's Longing we learn how she caught her ex-boyfriend and ex-work colleague and could hear them making fun of her figure, etc. I think that no small part of Lilli's self-healing process will focus on her finally managing to realize that her body is absolutely fine and she finally accepts it.
In conclusion, I think Lilith manages to bring a certain amount of drama to the plot with her storyline without letting it degenerate into trash on a Rosamunde Pilcher level. And while I agree, yes Lilith is very intense in the beginning, and I don't know how common it is for stepparents and stepchildren to have a strained relationship, at least in fiction it's a relatively common motif.