Translation:This highway goes through the whole town.
This is not how we usually say.
«Сквозь» emphasises the 3D nature of a city, it's not just a 2D path between the buildings. It's not just something around you, it's also above. This makes the highway look like a tunnel going through the city, piercing it. This might work for some science-fiction, or for some poetic descriptions, but we don't usually speak about highways in contemporary cities this way.
Why do you think it would?
There are no strict rules about the cases used with prepositions, such things need to be memorised on a case-by-case basis. But через is never used with genitive, it’s always used with accusative.
Sometimes, accusative looks like genitive. This usually happens animate nouns (that is, nouns that are used to name people, animals and other living creatures), but город is an inanimate noun.