The Asian and African elephants were probably safe being way further south so never ran out of food sources.
The gestation period for Asian elephants is 617 days and African elephants is 645 days, almost 2 years, so I think the mammoth would be similar. Whereas the gestation period for a bison is only 285 days, so they reproduce much faster...............
Elephants can be found living today as far north as the desert area of Mali. The lack of widespread grasslands there seems to be no encumbrance to their survival. Furthermore, Colombian Mammoths lived all over North America including Mexico, which is approximately the same latitude.
I find it simply inconceivable that grasslands would have been so greatly diminished due to any global warming period around that time as to have caused the complete extinction of such a wide ranging species.