Uh...I don't think your experience, motivations or family situation are comparable, Gotou.
And damn. Imagine what a spit to the face hearing that would feel like to somebody who didn't get the SSR gacha pull of a standup teacher with a supportive, understanding family being the first and only ones to take you in (entertaining your impulsive act against boredom for an entire month even despite the possibility of catching a kidnapping charge).
It's not a competition, Gotou's using her past experiences to find some common ground with Sayu, then they empathize with eachoter and Sayu will lister to what Gotou has to say.
It's a common strategy to make it so that kids and teens listen to you, show them that you are capable of empathy and you can somewhat understand them (Even if you can't understand them, but you know better, you're an adult and they need guidance).
And it seems like Gotou hit the nail in the head with Sayu trying to run away from life and from what she is. But because of reasons and personal philosophies discussed in the last like 40 chapters of this manga, we know that Sayu can't run away from her past and she needs to face it.
Also, the reason for Gotou to be acting strange with Yoshida may not have anything to do with her past experiences or anything like that. Maybe that's just her personality and she can be both
that kind of woman and empathic to the feelings of a lost highschool girl, because real people's morality is not a RPG table but a fuck-ton of shades of grey.