They have to breed the way they do to maintain the populations they had / have. Now along comes whitey with his medicine (and, notes the otherwise horrible mortaltity rate there-in), and decieds to do something about it, and actually meets it with some success.
Are you aware of how long it took for the exact same medical methods, etc, to bring the European and European settler state birth rates down? At least three generations, before the average woman decided that birth control was reliable enough, that obstetrics and gynecology was reliable enough, that early childhood medicine was reliable enough for her and her man to stop planning for at least 50% child mortality - JS Bach being a prime example.
And it won't go down if they aren't sure that most of their children will live.