Weedeater 500 B&S Engine only runs till Prime gas used up, New carb same result

I have 5 year old Weedeater 500 with a Briggs & Stratton engine. When I went to use this mower last week I encountered a problem After priming, engine starts right up but dies as soon as priming gas used up. At the end of last season's mowing, I ran the mower till the engine died and then tilted the mower to empty out the remaining gas . This year after fresh oil change and gas filled, I encountered this issue immediately on attempting to start. After trying a new sparkplug, checking the air filter, I came to the conclusion that the carb might need rebuilt. I just happened to have a new carb that I installed but surprisingly encountered the exact same problem afterwards... Engine starts immediately but dies after priming gas used up. On searching another forum I came across a similar report from several years ago of the same issue with a Weedeater brand lawnmower, which that person indicates they were able to fix. I wanted to ask if anyone can confirm/explain if what this person indicates solved his problem: by removing the safety-lever/brake mechanism, i.e. he removed the wire-bale on the handle that you have to hold down when starting/running the mower. Aside from the safety implications, I was not aware that the wire-safety handle brake had any function other than to act as flywheel-brake to stop the running engine. Is there some other mechanism at work with the flywheel brake? I remember from the old lawnmowers where there was a Slow-to-Fast throttle control mounted on one side of the handle and that the control lever had an OFF or STOP setting that stopped the engine when you pushed it all the way to the OFF setting. Does the flywheel-brake mechanism have any such function that might explain Why the engine won't run - though that totally ignores why the engine will start with priming gas...? Something else I came across was that the priming gas mechanism operates differently than the normal running-gas mechanism when the engine is running normally. And it struck me that I really don't understand how these B&S carburetors work. I'm assuming it's using engine compression to create vacuum pressure in the carb when you pull the cord, to pull gas up through the long narrow tube that 'sits' inside the gas tank and then to transfer the gas to the larger diameter but shorter cylindrical tube that's covered by a wire-mesh screen. From here it looks like the gas is routed to the intake-tube for the combustion cylinder. That's the limit to how I think the carbs are working but appreciate anyone's better explanation if not complete or...