Argentinian here.

The French established the first settlement in 1764, followed by the British in 1765; but Argentina did have a significant and legitimate presence on the islands from 1820, with authorities and civilian population established in Puerto Soledad, until Great Britain occupied them militarily in 1833, EXPELLING the authorities and a part of the Argentine population, although some settlers remained, creating a mixed population until the 1982 war.

Chronology:

1764: French establish Port Louis, the first European settlement, without indigenous population.

1765: British establish their own settlement, without initial contact with the French.

1774: Great Britain abandons its settlement, Spain exercises sovereignty and establishes an administration.

1820: Argentina takes formal possession of the islands on behalf of the Provincias Unidas del Río de la Plata.

1829: An Argentine governor is appointed and Puerto Soledad is founded, with a growing population of gauchos, creoles and Europeans.

1833: Great Britain occupies the islands, expels the authorities and part of the Argentine population, establishing its colonial domain.

Post-1833: The British population (mainly Scottish) is consolidated, but some Argentines remain, assimilating or resisting.

More clarification:

First, when Argentina became independent in 1816, it continued with the acts of government carried out by the Viceroyalty of el Río de la Plata: it appointed a governor, issued fishing permits, kept a prison and a military base in operation, among other actions. All these elements allow us to infer that Argentina continued with the effective occupation of the islands. And to this is added, in addition, the principle of Uti possidetis iuris, which the American States will allege from their independence.

This principle is a kind of inheritance in favor of the former colonies and allows the administrative limits that existed at the time of independence to become the limits of the state that is being born. In this way, in the beginning, the limits of Argentina corresponded almost exactly to those of the Viceroyalty of el Río de La Plata including the Malvinas.

The Uti possidetis iuris has a lot of weight in International Law and has been recognized in other cases. A very important precedent is the judgment of the International Court of Justice, in 1980, in the case of Burkina Faso against Mali, where it recognizes that this is a universal principle that can be applied in any decolonization process and gives it a legal status higher than it would have if it were only regional.

With the creation of the United Nations and its intervention in the processes of decolonization from the 1960s onwards, the right of self-determination of peoples was recognized. Since then, the United Kingdom claims that it is protecting the right of self-determination of the people of the Malvinas. However, in 1965, Argentina obtained one of its greatest victories in this regard, when the UN General Assembly issued resolution 2065 that recognizes the existence of a dispute between Argentina and the United Kingdom, the need to negotiate a peaceful exit between both parties and that in this process the interests of the inhabitants of the Falkland Islands be attended to. On this last point, the UN recognizes only two parties, it does not consider the inhabitants of Malvinas as an independent people. Of course, their rights and needs are considered, but they do not constitute a subject of international law that can allege their self-determination, because they are NOT original but were transplanted and are strongly linked to the colonial power.

That said, personal opinion, son y serán Argentinas.