I had the same problem for a few years. Finally, when I retired and got time to tinker with it, I ended up gutting the vacuum powered fuel pump and replacing it with a generic electric powered replacement. Problem solved. Starts first time, every time.
I still have the Snapper 30" RER that my dad bought around 1968. Am on about the 8th or so engine on it. Currently have a Briggs & Stratton 12.5 OHV replacement engine on it that I got from Smallenginewarehouse.com. Has plenty of power and starts very easily with the electric starter. Will...