Big flightless birds never adapted to the cold weather unless they were exceptional swimmers [penguins].
S. America and Africa have large flightless birds [emu, ostrich] and before Australia [elephant bird].But Eurasia and N. America only had large flightless birds periodically.
Not saying they are incapable of adapting to these environments but before the massive continental collisions around 30 million years ago, megafauna near the equator had big flightless birds and mammals [both herbivorous and carnivorous]. North of that megafauna was almost exclusively mammalian with the only birds being flyers or swimmers.
When the continents collided you had a massive life exchange and most big birds went extinct [likely due to their size constraint [due to their stance, that is another topic]]. A size constraint led to their incapability of competing with large mammalian carnivores/herbivores unless they could run away or fly away.
A few big birds made it northward and had success but it was short lived [3 million years] because the iceage wiped those out [as it did many mammals]. Not saying big birds like the ostrich can't adapt The only big flightless birds that survived continental collision were fast runners. Australian elephant birds survived because they NEVER collided with another continent [they recently went extinct when humans arrived], New Zealand and islands have several flightless birds which survive today because they are also isolated.
Birds are the most successful vertebrates on land today [by far most species], they are the only relatives of the most successful vertebrates to ever walk the earth [the dinosaurs] but they originally evolved for flight and are incapable of competing with largest placental mammalian carnivores or herbivores unless they can fly/run/swim away.