Ice age was not that much colder than now. Most of Canada was covered with ice, but most of United States wasn’t. Places which are hot today were warm at the time, and places that are warm were temperate. This means that there is no reason to assume that food was scarce.
Here is a way to think about it: Alaska today is probably colder than southeastern US was during ice age. Is food in Alaska so scarce today that it does not support megafauna? Obviously it’s not, quite the opposite. Same was true about most of US during last ice age.