They need to do both:
Drive a big truck around the city streets pushing a plow to clear them of snow and ice
And
Drive a big truck around to collect garbage
It's much more efficient to do both at the same time, use the same trucks and skilled labor to run them.
It's a classic two birds one stone situation
You can just go on, making it less and less efficient each step.
Point is, a truck that plows snow and a truck that is a trash truck are mostly the same thing. You take a diesel cab truck chassis, and add either a plow or a trash compactor to it. It's also easy to add both.
It's not as if the trash trucks in cities that don't use them to plow are some highly specialized thing. They're the same trucks as new york has, just no plow. A plow bolts on.
Still get the economy of not having to buy twice as many big trucks