Unless you have a very low pressure gauge, you most likely will not get a reading when dead heading.
Fuel pressure and carb engines is going to be 7 PSI or a bit lower.
Yes, starter button will typically over ride "safeties" like fuel pump and oil pressure cut out but that is temporary at best and some advanced safety systems may even limit the time they are over ridden..
Spraying starter fluid into carb and the engine continues for a few more seconds is a pretty big clue as to the problem..
Engine is starving for fuel, you have proved the fuel pump is pumping fuel, you sprayed starting fluid into carb and engine continues a bit longer..
Basically the troubleshooting done is pointing to the carb as the culprit.
Carbs get gummed up, floats get stuck, dirt clogs the jet and your carb needs serviced or replaced.
By the way, instead of starting fluid (Ether in a can), you can substitute spray carb cleaner like Gummout.. , Just spray in like starting fluid and will act as a burnable fuel for a few seconds. Much, much easier on engines that Ether based starting fluids..