As per your suggestion, I gave your article a read, and to be honest I notice a few flawed arguments.

For one thing in contract law, you don't need to sign a contract for it to apply. Rather in a social contract the contract is ratified by assentment.

The moment you "sign" the contract is when you pay your taxes, which children also pay in the form of transactional and income taxes. And each time they do that, they are in fact partaking in a contractual agreement with the vendor. Such as buying sweets or school lunches or whatever.

So minors can actually participate in lots of different contractual agreements, and that's a good thing because it provides contractual obligations intended to protect their rights. If children were completely excluded from contractual law, then they would have far fewer consumer rights.

Moreover, parents can also enter minors into contractual obligations, most common being that between a landlord and a tenant for instance, wherein the landlord is just as liable for any misconduct or negligence caused towards the children as that of any other household member.

So if you concede that minors cannot sign contracts, then you are using contractual law as a framework, and since contractual law permits contractual agreements by assenment, then you must therefore concede social contracts to be valid.

But that's a legal argument. That might work in a courtroom. You also made a philosophical argument. Two in fact.

First one was "Why do you have the right to apply these laws?" To which you concluded that the only justification, or at least the primary justification, is force.

But the problem here is that this inquiry is a self-defeating premise, because force is not actually a physical phenomenon. Rather it is a framing of intent regarding physical phenomenons. Which makes the argument metaphysical in nature.

And then you can examine this intent quantitatively, and conclude that the only reason the force is applicable is due to how it is produced by a collective intent which outnumbers the individual human.

And since human societies exist within a eusocial division of labour and social organising and we are in fact not asocial animals, then there is a scientific justification in saying that the combined intent of others outweighs the value of the contradicting intent of the indvidual.

Because it is precisely like you concluded, that it is "Because we said so." That's the key word here. Another vital part is also the saying part. Our capacity for complex communication and larger scale social organising actually puts a lot of value on our capacity to assert our will through cautionary language instead of outright force.

So the scientific justification could just as easily be "Because we are capable of doing so."

And moreover I am not sure there is much evidence for our abilities of voluntary organising. Because if we were capable of this, then why do psychological phenomenons such as culture, customs and language exist? Our sensory physiology with regards to neurological and organic forms show that we have sensory externality or continuity that penalises individuation.

In fact, isolating ourselves and refraining from communication and social organising causes us physical and psychological detriment in the form of nerve and brain damage. As seen in people who are kept in solitary confinement.

So the scientific justification is that we have a biological imperative to large scale social organising as per the Maslowian basis.

And as such the further justification would be "Because it happened."

But even then that is, like all scientific justifications, entirely selective. Because one could argue a larger metaphysical basis on the etymological grounds of logical positivism, and simply proclaim that there is no fundamental evidence to suggest that, objectively speaking, a state mechanism is justified.

But the problem here is that we don't exist in an objective ether, and since you yourself denied religiousity, who can mediate such evidence? To what authority do you submit such evidence?

The other option is a secular authority, such as a courtroom, but in order to have one of those you need a state mechanism.

And so, in asking why the state mechanism is justified, you are entering into a paradox, since it is only a state mechanism which would be able to hear such a question and exact a meaningful verdict to such a question.

The moment you are proven right, your evidence lapses as the authority which recognises such evidence lapses too, leaving room for the whole process to repeat again.