True, meatpacking doesn't requires a lot of workers. But agriculture does, and, inevitably, at some point the British would try tu use these lands for growing wheat, as Argentina did in the late 1870s. And that's when immigrants woulkd be needed, and in great numbers (specially since the local gauchos weren't particularly willing to work in the fields growing and harvesting crops).
Also, the lands were occupied, but much of the pampas, Chaco and Patagonia were only occupied by Nomad Indians, who wouldn't have been considered "owners" of the land in those days.
This said, I understand your point: these lands weren't as empty as, let's say, OTL settler colonies as Australia or Canada. Thruthly, those weren't empty either, but were sparsely populated by huntergatherers whose culture was very different from European one. Argentina had much more people (between 500,000 and 700,000 according to the source), who spoke and European language, and who practised a religion practised in Europe.
In no realistic scenario you can avoid these people from having a distinctive impact in the culture of the country, even if hundreds of immigrants do come.