Certainly, much human evolution happened before the ice age. Behavioural modernity, though, might have resulted from a shift in software rather than hardware, due to the advent of culture (the externalization of the mind in symbols). I'd emphasize a tantalyzing existential reckoning that the ice age would have facilitated. The environment's transformation due to climate change would have called attention to its independence from the mind, resulting in anxiety and alienation, and in the realization that humans are essentially at war with an environment that doesn't always have our back. We had to get smart in building an alternative, artificial environment, one less indifferent to our welfare, and the colder ambient temperatures would have supported that ingenuity by encouraging cognitive experimentation.

Once the temperatures warmed and the ground thawed, at the end of the ice age, we were ready to put those newfound convictions into practice: hence, agriculture begins at that that point, followed relatively swiftly by civilization. The ice age was like a giant time-out, during which we stewed and contemplated our existential condition, and that antagonism towards nature looks like the hallmark of behavioural modernity since it's what motivates the use of culture/symbols to transform the wilderness.