Copa America doesn't generally involve lots of traveling from fans all over the continent. There's no tradition of that, in part because it' been played so often. It's not only that distances are big; airfare in South America is also very expensive and most people don't have a ton of money; there's no trains; and buses take forever and in many places are pretty dangerous. Most people are also not in long holidays in June, as the summer is December-March. World Cup is a once in a lifetime thing for most people so it's a whole different issue. I've traveled to the World Cup in Russia, but have only attended Copa América when it was in the city where I was living.
It's usually the locals who fill the stadiums in most Copa América games, with some help from immigrant communities and some travel from those close to the borders, plus the odd few thousands of hardcore fans who will travel anywhere. In some cases this works well anyway because the games are in cities that rarely have international football, and the stadiums are not that big anyway -- so you'll see 25,000 people in a second-tier Ecuadorian stadium, which is enough to almost fill most of them or at least look good.
In Brazil many cities have a large offer of football, and the stadiums are big. In other words, you don't expect 30,000 Paraguayans to travel to watch a Copa América game against Qatar, and it's hard to imagine that 80,000 people in Rio de Janeiro will fight rush hour traffic and spend money to attend Maracana for that game, or same in Sao Paulo to fill Morumbi for Chile - Japan (although there are a lot of people of Japanese origin there, most are Brazilians who support Brazil) when on paper they can watch better football, with teams they care about, every weekend. So in fact around 20,000 people for those games is not bad.
By contrast, games in the (very long) WC qualis tend to be packed or close to capacity as long as the teams have a fighting chance.
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