I don't think the formation of carbonic acid has much to do with this. While the formation of carbonic acid is a slow reaction, the reverse reaction is quite fast. This means that CO2 will be available quickly whether its already CO2 or still in its hydrated H2CO3 form. Further more, the amount of CO2 compared to H2CO3 is quite large. When equilibrium is reached, there is about 600 times the amount of CO2 compared to the amount of carbonic acid. As a result, the carbonic acid should play only a minor role.