Would life come to a standstill for days or weeks, or at least limit mobility dramatically, as people waited for the accumulated snow to melt? Or were brigades of shovel-carrying laborers and horse-drawn plows on hand to clear streets and roads?
I started writing an answer last night but thought that 1. St. John's Newfoundland is barely a city now and was just a big town at the time of my sources and 2. my sources are from the 1880's so not sure if that is "premodern". However, without an answer other than the link I thought why not?
I think that to properly understand what cities did with snow you have to think about how people got around. Before the car or when street cars glided the streets people got around either by foot, or by being pulled by horse or dog. With that in mind the vehicles that were pulled didn't need to have wheels and clearing streets down to the ground wasn't practical. If there wasn't a huge amount of snow people just used sleighs, eventually the snow would be packed down and it was basically like having a road (based on the descriptions of the roads I have seen it might have been better) and carriages could be used again. Now if there was a dump of snow people would be employed to deal with the drifts, on January 10, 1879 poor old John Donovan, an employee of the Municipal Council of St. John's was engaged clearing Harvey's Road for traffic from the Tabernacle to the Parade Rink. On January 27, 1888 the community of Bonavista reported that they were in "...the midst of a cantankerous, old-fashioned Newfoundland winter." which is the best description of a Newfoundland winter. They reported that the snow made travel difficult if not impossible and men were shovelling snow to make thoroughfare passable. A description of the specifics of winter road clearing is recorded in the proceeding of the Legislative Assembly when they were debating the municipal act of St. John's. The Hon. Mr. Morris stated in 1890 "In the summer season there must not be a rut in the roads around the city, or an unpaved street; in the winter the roads must be cleared of snow and the gulches (pot holes) levelled, and all this for the accommodation of those who have carriages and sleighs."
Oddly enough the speech from Morris wasn't as much about vehicles but about pedestrians. Now, as a personal note anyone who has spent a winter in St. John's knows the plight of the pedestrian. When a normal winter sees about 300 cm there is a lot of snow to move and clear, sidewalks are some of the most difficult to keep clear this is as true today as it was in the past. An editorial from February 1882 wrote to bring attention to the "honourable" board of works on the snowdrifts on Water Street causing pedestrians to walk in the road and "...thus expose themselves to the danger of being run over by passing horses and vehicles.". The editorial continues that "It will be observed that the authorities, or their officials, are very prompt in compelling Merchants and Shopkeepers to keep the sidewalks in front of their respective establishments regularly shovelled, and the snow levelled upon the roads..." And yes! Sidewalks were to be cleared but the snow was to be thrown in the road to level it. Police even enforced this rule, from February 1880 "a policeman called on the various dwellings and ordered that all snow lying before their doors should be thrown into the middle of the street.". The clearing of sidewalks led to horses being ridden on the sidewalks, this dangerous practice soon lead to fines, like John Neville, 30 from Water Street who was driving his horse along the sidewalk near Patrick Street. Mr. Neville was fined $1 plus cost on February 6, 1882.
I hope this helps answer some of your questions but the long and short of it is normally people just rode on the snow. If there was a large storm things then much like today would shut down for a time and then the roads would be levelled and life would return to normal.
Sources: All from The Evening Telegram, in order January 11, 1897 January 27, 1888 May 13, 1890 February 11, 1882 February 6, 1882.
Off topic slightly, but in 19th and early 20th century New England, they used horse-drawn snow rollers to pack the snow down. Packed snow was easier and more consistent for sleighs and sleds. And essentially every type of wagon also had a sleigh version (there were even hearse-sleighs for conveying the deceased).
I do not know if this idea came from older times and technology in Europe or Scandinavia, or was a Yankee invention.