I assume if it's vital infrastructure to a couple mega-farms they'll happily take over maintenance. They're not gonna just shut down operations because some rural bridges on their preferred routes aren't maintained anymore. They'll either accept longer routes using a smaller set of still-publicly-maintained bridges, or solve the problem some other way, maybe by maintaining the roads themselves. Public spending drops, and one or both of food prices go up slightly and profits go down slightly. Seems fine. May well be cheaper, overall—forcing companies to choose between longer routes and maintaining their own roads seems like a good way to weed out routes that are being maintained at a net loss.
Why do folks act like farmers are doing what they do out of altruism or something? "You owe farmers! They feed you!" Yeah, I do owe them. I owe them what I paid for the food. Right? If they decide I'm not sufficiently grateful they can stop. I'm not a "the market solves all problems" sort but I'm pretty sure it'd solve that one, no problem. Others will happily increase production to make up for the loss, because it's a market. Throw a fit that urbanites aren't subsidizing rural living & businesses enough, and stop producing, and someone with more sense will gladly take that market share.