If Brazil has a radar facility that receives Mode C transponder data why wouldn't they have a VHF transceiver at the same site so they could talk to them since they both operate with line of sight frequencies? Are they sure the transponder failed on Legacy when ATC lost it or were they out of range in one of their blind areas. Since ATC didn't question the loss of Mode C they must have experienced it routinely so didn't question them. Why would software in their radar room automatically display them at their flight planned altitude because they lost Mode C? The shift change might have contributed to why them reporting in at FL370 and later displaying FL360 didn't get picked up.
I think Brazil has to fix their ATC problems quickly because blaming the American pilots isn't going to bail them out.