The result would be the opposite, due to how lopsided the urban/rural divide is in the US. All politicians would have to care about are a handful of large cities, ignoring the rest of the states.
One of the troubles with this argument is that it fundamentally assumes that people who live in cities don't care about people who live in rural areas. People are capable of having empathy and understanding of people who are not like themselves. We don't need to Other/make outsiders out of fellow citizens who have different lifestyles than us.
The better solution would be something like separate elected offices for different interests, with representation based on how much an area is affected by that legislation, instead of a single position for everything.