No, you haven't missed anything. The family has no connection to Patagonia except as a symbol. Flying the flag of the "Kingdom of Patagonia" is used by the far right in France in a symbolic way. There is not much information readily available online, but here is what I have pieced together. In the mid-1800s a French adventurer declared himself King of Araucania and Patagonia but the Kingdom was never a diplomatically recognized state. The indigineous Mapuche peoples in Patagonia opposed to colonial rule went along with the adventurer's claim; the speculation I read is they may have believed having a European as the head of the state could help them regain autonomy, that the colonial powers might accept an independent state with a European as king.
en.wikipedia.org
The families who were renovating Boulard into their summer paradise in the 1970s flew the flag of the Kingdom of Patagonia over the hamlet. That's the reason Patagonia comes up in articles about Emile's disappearance. French novelist Jean Raspail declared himself consul of the Kingdom of Patagonia. In the 1990s, Raspail and a few others twice attacked a deserted island called Les Minquiers, which is south of Jersey and under British control. He took down the Union Jack and hoisted the flag of the Kingdom of Patagonia.
The families summering in Boulard flew that flag of Patagonia as a declaration of their shared worldview. Raspail's 1973 novel "The Camp of Saints" depicts France and the West being overrun by hordes of non-Europeans. It is called "the racist anti-immigrant bible of the extreme right" in the article above; Raspail came to be considered the Prophet of far right extremism. The homage to Raspail in flying the flag over their summer sanctuary indicates support for extremist action and opposition to democracy and a rigid, patriarchal family life.
From the very limited information we have access to, the family's rigid beliefs and actions re democracy or immigration may not be as relevant to Emile's disappearance (although authorities are investigating all avenues) as what it suggests about their views on childrearing and strict discipline. That's my opinion. I fear that a little boy escaping out the gate on his first afternoon with his grandparents may have needed to "learn a lesson" on disobedience. MOO.