Argentina was kicked by Spanish and British policy, with both at different points in the Early history of Argentina dominating their economic institutions, effectively draining surplus value. Development was uneven and often under control of either foreign backers or the elites of Buenos Aires. The export booms which got it to 1st world standing by the 1920s were not enough. Internal growth is needed to keep Argentina from shuttering every time commodity prices drop.
part of the US advantage is the massive unbroken farmable tracts from Ohio to Colorado. Though many immigrants kept close to the East, the westward conquest and the cheap, US army secured land was a real big attraction. Ultimately the US just has a land advantage once they yoink Louisiana. Most of northern Argentina is pretty good land, but the entire country could fit in the Mississippi drainage basin.
That said, if Argentina can avoid the allure of easy money with Britain (say relations stay sour) they can start the process of selling tracts of land to grow bases of power away from Buenos Aires and the free trade regime they favored.
Let's say an Argentine independence happens with lots of hostility towards the British. The new Argentine government, though not hostile to Buenos Aires, decides against a capital so close to the sea and chooses to build a new capital inland near Rosario (Let's call it Ciudad Argentina). Pushing political and economic power away from Buenos Aires would do Argentina a lot of good, and the government actively encourages settlement of contested areas with settlers loyal to Ciudad Argentina. The government invests in building a rural middle class to counter both urban elites and build a tax base for levying armies and building infrastructure. Investment in agriculture and the building of a large middle class snowball, as more immigrants have more opportunities and further invest in a government pushing for immigration. A virtuous cycle kicks in and this Argentina has a lot more luck building itself into a US lite.
As for discouraging immigration to the US, there's loads we can do. Have the Continental Army move against the Continental Congress in early stages and mess up US institutions from the get go, meaning more instability and unrest, etc. Keep Articles of Confederation, have a more Federalist and exclusive Constitution, etc. Etc. Basically keep US institutions either unstable and unpredictable or exclusive and otherwise unwelcoming