Hi Swiftdevil and Mod Gadge,
Your problem is clearly the fuel pump diaphragm which is the very thin black rubber type item that is clearly fuel affected and thus is not allowing the carby to pick up fuel from the tank.
This is the most common issue with most Briggs engines of that era.
If you follow Mod Gadge's advise you can't go wrong and you'll have it up and running in no time. I will say though that these small parts are not cheap so be prepared for a bit of a shock on that part as they are around $11 ~ $15 depending upon where you purchase it from.
Cheers,
BB
