Search flights from entire Europe to entire SouthAmerica in a single search?

The website http://matrix.itasoftware.com/ (owned by Google) supports this a bit, but you need to learn how to use it for complicated question like this.
In a nutshell, you enter a city pair from about the middle of your starting area to about the middle of your target area, and then choose to allow 'all airports within x miles' (1000 or whatever you want). In the offered list, you can select all airports you want to include (or simply select all). You can then chose to search over a month, and sort by price.

There are some limitations - your starting airports must be all in one country is the worst one; it limits the usefulness for your case a lot.

For Feb 2021, I found FRA-GRU for 309 Euro as the cheapest (multiple days, for example Feb/10)

Which sites exist ?

Quite likely none since it's a non-sensical search. There are 420 major/medium airports in South America and 1026 in Europe, which creates over 42000 possible combinations. I imagine you can hit pretty much all of those with two connections and maybe 25% with a single connection. That's still over 10,000.

It's just not practical to search all of these comprehensively and display them in a way that makes sense.

What would you do with that amount of data anyway? You have to have to some reason for looking and that reason should help constrain the search to something more meaningful. That's where ITA Matrix is probably the best tool for.

Kiwi.com allows you to execute a search like this but I sincerely doubt that they search 42000+ connections. They simply will constrain the search for you by some unknown algorithm or rationale. You are better off constraining the search yourself.

However, the combination might work: Using KIWI to pre-screen options and then refining the with ITA. One word of caution: KIWI offers frequently itineraries that are on two separate tickets and require "self connection". You won't find these on any other website: it's risky business and unless you are an experience traveler and know what you are doing you should NOT book these.

I found https://panflights.com/en/ Seems to be interesting although not perfect and with all functionalities needed to be a flexible search engine