Argentine peope are not a race and I am surprised that has to be pointed out.
That said.
Argie is a word I believe originated in England, although I stand to be corrected on that. It is used in both a derogatory way and an affiacionate way, depending on the intent. It too should not be used and I personally do not use it and bristle when people in the UK do. However, there are some key differences between this and negrito and chinito. The first and most glaring (which is why Argie should never been offered as a counter in the first place) is that Argie is not used to describe all people from the Latin American countries, like Chinito in Argentina means anyone from a certain region, regardless of their affiliation with China. Sure, Argie is offensive to a group of people and I absolutely disagree with its use, but it is not lumping several nationalities into a single term. Similalry negrito, which as I pointed out is also used to describe my nephew who is not even black. But hey, you have dark skin you're a negrito.
Put it like this, whether the word is horrible or not, you are an Argie as that word describes people from Argentina (sadly). A Japanese person, for example, is not a chinito.
Like I said, when people used Paki in the UK, the truly discriminatory thing about it was its use to describe all people from a region because of how they looked and to verbally segregate them. By the way, we are all not discussing the fact that Japenese (for example) people actually don't look like Chinese people, which makes the term chinito even more offensive.