Most true suburbs aren't within the big city limits, so I'm not sure your point is well-founded. For example, in the DC area, the suburbs aren't even in the same state as the city and yet the suburbs seem to be thriving.
DC is an exception, most American cities have large swaths of suburbia within the city limits and even larger ones within the same state.
Those suburbs outside the city limits still need money. They get it from state and federal funds which were mostly collected from people in the city limits. For example the Highway Trust Fund as one of many examples. If you check the per capita spending, it's higher for suburbanites than urbanites but the urbanites are putting in more money.