From some quick digging it appears that people normally just "dug in" when large snowfalls occurred early on. Horse-drawn plows were at the top of the technology tree for effectively plowing snow until modern plows came around with automobiles. Some carriages and carts were even made into sleighs. That is, if the horses could even travel in the snow. "The Great Snow" storm in 1717 which took place in the area you're asking about pretty much made travel impossible between Boston and New York according to the NSIDC. That article pretty much covers how we adapted from early to modern snow removal.

Interestingly enough: avoiding the snow-piled streets was also one of the many motivations for some cities to build subways. Almost in effort to just avoid the problem of removing it in my opinion!