What makes it dumb is that the clothing described wouldn't have been what kept them warm and unfazed, that would be the layers underneath the made up cowboy's attire.
What makes it silly is that cowboys exist across the very snowy north and have since cowboys have existed anywhere.
Not American but doesn't central America get cold as fuck at night? I would assume that's what cowboys dressed for since you can always remove clothes but you can't exactly create them from thin air at night.
I'm not an expert, but I believe it's more of a North American thing (Canada, U.S.A., Mexico) due to the mountain systems along the three countries. The Rocky Mountains, the Trans-Mexican Volcanic Velt, etc. are all part of the North American Cordillera. This, and the occasional deserts.
I'd guess ponchos, jorongos, and similar pieces of clothing were adopted by non-native settlers (Spanish, English, etc.), including non-native cowboys, because they are good against the changing weather during the day and the cold nights, as you said.
I mean, Central America must have cold spots along their own mountains and South America has the Andean Mountain Range (enormous system), but I do not know about their traditional clothing, except they share the poncho, and I do not know which of their clothing we still wear to this day.