Slave- noun
(especially in the past) a person who is the legal property of another and is forced to obey them: he killed the natives or turned them into slaves
a person who works very hard without proper remuneration or appreciation:by the time I was ten, I had become her slave, doing all the housework
a person who is excessively dependent upon or controlled by something: the poorest people of the world are slaves to the banks she was no slave to fashion
a device, or part of one, directly controlled by another
An Online dictionary definition.

Berk doesn't have slaves.
It is rather proud of the fact.

Yes, it may snow nine months of the year, and hail the other three. Yes, some of them may starve in the winter when it is too stormy to fish and the dragons have stolen too many of their sheep. Yes, people may die when their leg goes green and smelly and they have to cut it off. Yes, Hiccup's generation may have had four more children in it that never made it to the toddling stage, let alone being any use to the tribe.

But Berk is advanced enough, sophisticated enough that it doesn't have slaves.
Right. (Everyone ignores the little niggling voice that whispers "What is Hiccup then?")

He helps out in the forge. Everybody knows that. Well, he does now.
It was a bit of a shame he couldn't be trusted to play outside in the fresh air like the other children did.
He always got too close to the cliffs, too curious about the lake, too interested in the flickering fire for his own good.
So keeping the boy indoors was only the sensible thing to do (and the chains only to stop him from running off, not escaping, of course not. He should have listened when Stoick told him not to run away to play with the other children near the firepit).

He doesn't get paid for his work.
That's true, but he's only working for his keep (for his bowl of luke-warm stew at the end of the day, for his place far away from the fire and fellowship).
It's not as if it's very hard work (Gobber only lost his hand to it, Little Hiccup's arms are only shivering from exhaustion at the end of the day, he only has a few hundred scars from the sharp swords and burns from hot metal on his body) after all he, the failure of a Viking manages it.

In fact, only thing that keeps him in the tribe is that he is the boss's son (although even that fact wondered about behind the Chief's back, but not to far from Hiccup's ears, "it was a very long time that Stoick was away on that voyage, if you catch my meaning") and that no-one wants to approach Stoick about finding an alternative heir (even if he is a useless boy, a waste of space, surplus to requirement – after all who needs a Viking that can't even kill a dragon).

But still he's part of the tribe, he still belongs to it. And he will do what the tribe wills or suffer the consequences.
His little toys, his precious "inventions" as he likes to call them, are occasionally useful but not worth the constant disobedience and unVikingness the child has.
After he is declared an adult, after he has finished Dragon Training – as if that little wimp will ever pass it – he will have to conform or pay the price.
After all he is expendable (he's just a slave).