While the Federation appears to be based on a form of American government, lets not forget that while the US government has looked the same from the outside since it was created (President and his cabinet, Congress) that the power of the Federal government was limited compared to today. It evolved into what it is today, and while the Federation government is probably doing the same in the Star Trek verse, we can still have a limited government with what we saw.
The member worlds can still have a lot of power, and the Federation may provide support to these worlds power. Todays example is taxes. Wealthier US states pay more taxes and are a ble to self sustain in a lot of areas, while poorer US states don't pay as much and receive Federal aid to help the states stay equal with the others.
So for example, a planet like Bajor is receiving a lot of Federation aid to build the planet up. On the other hand, Vulcan probably receives little aid since it doesn't require it. As long as the Federation has a large amount of resources in its control, it can redistribute these resources to the poorer planets to help encourage development. If this can be sustained (the poorer planets building up and becoming suppliers themselves) then the Federation can continue to expand.