Her relationship with her parents is exactly what made her reject the idea of ever having kids of her own.

Okay, Briar knows what her parents faults were and what her own faults are (unlike them, who didn't want to admit being in the wrong), but she can't help but feel like she will still find a way to fuck it up and would be a shitty parent even if it's not her intention.

Briar knows she is prone to starting arguments, and she would be afraid of raising a kid who's afraid of talking to her in fear of getting scolded or who knows what, just like her mother did. And she knows she's selfish and she admits it, so what if she puts her own safety first instead of her kid's, just like her dad did?

I think Briar has been told so many times in her life how similar she is to her parents (her mother would always tell her "she was just like her dad, only looking after herself" when Briar tried to defend herself; and her father would tell her "she was just like her mother, always finding a reason to fight") that she's afraid she's the same as them.

So yeah, the fear of becoming the same parent her parents were (even if she recognizes her flaws) and doing the same mistakes is what made Briar decide to never have kids.

And another reason she has that prevents her from ever wanting a kid is knowing how her reputation would affect them. Briar was the child of two normal people (her father was a punk musician, yes, but he wasn't world-famous; and her mother was just his high school sweetheart), and yet she grew up with so much weight on her shoulders to make the family proud. Now, how would the child of the Star Supernova, the incredibly popular idol and also genius scientist who researches Legendary Pokémon, feel? She knows her kid would experience far worse pressure than she ever did, and she's not heartless, even if she tries so hard to pretend she is. She can't do that to a kid.