Quoting nineteenth century battles as a reason to override the wishes of the residents of the Malvinas is silly.

The people who live there get a say, and no one else. The Battle of Altos del Quebracho (1846) has literally zero relevance.

To be blunt, the UK and Argentina like to focus on the Falklands precisely because it's so meaningless, and allows the government to distract the populace with patriotic gesture and national pride, so they forget about how actually incompetent said government is. (And at this point, the UK has overtaken Argentina in incompetence - Bojo is worse than the worst of the Generals were.)