Most people don't need FFT algorithm for multiplying large numbers, Karatsuba's algorithm is fine. But in some domains the difference does matter.
Personally I usually see the opposite effect - people first reach for a too-naive approach and implement some O(n^2) algorithm where it wouldn't have even been more complex to implement something O(n) or O(n log n). And n is almost always small so it works fine, until it blows up spectacularly.