Responses
I guess he spied on them. Rember that Dabi was always gone or came later when the villains did their stuff? He said he tried to find new members but honestly what does he care about the league? I think he was absent because he spied on them or hired other villains to do so. Touya is obsessed with his Dad, I bet he's the stalker kind of guy who spends hours everyday to stay upgraded on his family.
You make a good point. He probably was spying on them especially since he was looking for opportunities to kill Shoto.
In chapter 290 pg 13, Dabi is talking about how Endeavor is trying to establish himself as a father figure and reestablish familial bonds with his family. But how exactly does he know that this is what Endeavor has been trying to do?
Dabi doesn't live with the family to see how Endeavor has been behaving at home and Shoto training with Endeavor wouldn't necessarily mean Endeavor is trying to change if Shoto is just a puppet/doll behaving at Endeavor's behest as Dabi believes. Additionally, Dabi wasn't at the Ending attack to see Endeavor hug Natsuo or hear what he said to Natsuo . . . so how does he know about this change of heart Endeavor has been displaying towards his family?