<<he'd switched off the transponder (for reasons unknown). That would have impeded the 737's TCAS from reacting to the Legacy's presence and delayed ATC's warning that a collision was imminent since, with the transponder off there would be a lag for radar to correctly track the Legacy.>>
There would be no "lag" that would of consequence if a return was visible. There are two types of radar but I do not know if Brazil ATC has both - Primary radar shows a response whether or not the a/c has a transponder and secondary shows coded returns from a/c transponders - usually displayed as a label beside the radar target of the aircraft. The display to the controller in each case is updated very frequently - certainly frequently enough for ATC to safely handle extremely busy terminal areas. If the ATC radar does not display primary returns then an a/c with no transponder will not be seen, ie if the pilot switches off his transponder the radar information is lost completely, although a trace of the a/c track could remain for a minute or so. In a positive control situation, if an aircraft disappears from radar for no reason then ATC will attempt to find out why.
As for pilots switching off transponders and changing their flight levels without ATC clearance... such utter irresponsiblity beggars belief.