Cheese???
So it's mostly just a source of fattiness and creaminess to add that velvety texture to the sauce
Maybe what's called 'cheddar' in the US is different too though, I don't know? It's not PDO (and the US doesn't respect them anyway I think?) but I'm comparing to British if not Cheddarian cheddar.
The non-'white' ones look like red leicester, which is a bit like a mild not very salty cheddar, and that colour.