The Argentine claim to the Malvinas has its holes for sure. I can admit that; the main one being that the settlement was in disarray by the time the British arrived.
I think the main ones are:
that the Argentine Convention was embroiled in a civil war with the city state of Buenos Aires in 1832, who had appointed Vernet, albeit without clear jurisdiction to do so. Britain was bringing rule of law to a troubled island occupied by the puppet of a city state almost 2000km away .
that 'Argentina' spent the next 50 years in civil war (Buenos Aires again) and beating up on Chile to acquire Patagonia - no problem with expansionism there.
and most importantly
that the 1850 Convention of Settlement restored 'perfect relations of friendship' between Britain and the Argentine Federation (again represented by the top dog from Buenos Aires) and completely failed to mention the Falklands, thereby acknowledging that prior events were rendered moot.
Perhaps this acquiesence of the status quo between 1849 and 1941 where no formal protests were made by Argentina to Britain is the fatal blow to their claims. International law seems to indicate 50 consecutive years without protest means all claims are dissolved, which they seemed to be in any case by the 1850 Convention of Settlement.