"a piping system can only supply as much as it's smallest diameter allows through"
No, it is not so simple. How much gas will flow through a given run of piping depends on lots of details. All piping adds 'resistance' to the flow of a gas or liquid. Pressure pushes the gas or liquid through the pipe, and overcomes the resistance. Smaller pipe has higher resistance to flow than larger pipe. Longer pipe has more resistance to flow than shorter pipe. A 90 deg. bend has higher resistance to flow than a 45 deg. bend, etc.
In the end, it is the total resistance between the source and the appliance that matters (as well as the available pressure). A short section of 3/4" pipe mixed in with a long run of 1" pipe will have significantly less resistance (and higher resulting flow volume) than the same total length of 3/4" pipe.