I believe if Freminet wasn't given to the Hearth then he would have been forcibly taken from his mother to be trafficked, when you look at his story it's the only conclusion that makes sense.
It's very important to remember Freminet was given to the Hearth when he was very young and still in his developmental years, “In the years that children from ordinary families ran about laughing, showing off the latest clockwork toys or picking up phrases from The Steambird, Freminet's heart was gradually becoming numb. Orders filled the void and became like a clockwork spring driving him forward.” He didn't get this mindset instantly so it's a safe guess he'd already been in the Hearth for about a year at this point making him even younger then the mentioned age range.
The reason Freminet was given over to the Fatui is because the debt collectors were trying to take him and his mother decided that that would be a worse fate than with the Hearth (It's possible she had a vague idea of what the Hearth was like because of the line “In the end, she had no choice but to entrust Freminet to "that orphanage" ” that orphanage clearly being a negative way of referring to it.) “ demanded that she hand over Freminet as well. But she was his mother — how could she let this happen? ” it specifically says that they demanded she handed him over, he who was still a child. Likely not even seven yet.
I also would like to point out the line “ "Still haven't figured it out? You were sold to settle a debt." ” While Freminet was being lied to the writers chose to say that he was sold to settle the debt, not given away or abandoned, they also chose to call the debt collectors “ Those greedy moneylenders ”.
