Everyone keeps saying how Lin is definitely Toph’s daughter because of how hardcore she is, but she is an individual too.
When Mr. Sato said he had no affiliation with Amon, Toph would’ve slammed her hand on the table(or as an adult, calmly placed it) and would be able to tell if he was lying.
Lin just accepted it.
Toph walked around with bare feet all the time (sometimes indiscreetly) so she wouldve noticed the tunnels under the house automatically, and the bombs in the arena.
When Lin was trying to persuade the council to keep the arena open, she used reason and hope in her soldiers. Toph’s got anger management issues and although has learned to become a team person by that age, is still most likely a solo act.