The father is illegal so if he showed up competition he’d be arrested and sent away too.
Not according to the articles I've read. The "father" (quoting because it's still unconfirmed) was supposedly a citizen. But with reporting as shitty as it is, this could be a bullshit "fact" just like the claim that the 2 year old citizen was "deported".
they were checking in to their immigrant center and going through the legal steps and still got taken away
ICE had final deportation orders... that usually means that the process was pushed to completion. Court case and all.

The child wasn't deported. The mother was, and the mother chose to bring her child with her. The child was not deported.