Honestly not the case where I am. State money only comes if for natural disasters: back when Hurricane Sandy came through, there was state money to replace bridges damaged by the rain volume. Otherwise it's handled at a county but primarily the town (which in NY is a "shire" or "borough" government level between county and city/village).

There's a rotation of road resurfacing that follows a 2-5 year cycle. The key thing with roads in snow country is getting the drainage right - as long as there's no standing or under-road water to freeze, the road won't get frost-heaves which are one of the primary damage sources. So our town crews focus on drainage first and then resurfacing, and then smaller bridges. Larger bridges are generally partly funded by the county and state because of the scope but the one bridge recently replaced was over 50 years old and the new one has a similar expected life.