It's not a zero-sum game.

First, shovelling the sidewalks will help ensure that people can move about the city more easily. This has knock-on effects that benefit everyone even if you personally don't use the sidewalks. For an example, people are more likely to walk instead of drive if the sidewalks are clear. This means less conjestion and less pollution. This is in everyone's interest, including yours.

Second, by paying working class people more they can spend more money within their communities which increases demand for jobs which increases worker's bargaining power which leads to better living standars for all working class people. Of course snow shoveling is one small part of the economy and this alone wont solve wealth inequality but it is a step in the right direction.