PTO problem with John Deere F525

After some years (5+) of totally reliable starting, my weed whip won't start. I found and replaced a couple of rotten fuel lines but still have the same symptoms. I notice that when I push on the primer bulb, it doesn't pop (or even ooze) back out. I sprayed a bit of carb cleaner into the carb and it'll run for a second or two before dying (obviously just burning up the carb cleaner) so I am guessing that it isn't pulling fuel. I probably have air in the lines (and don't know whether that is a problem). If I take the primer bulb off, it pops back out. I see a very thin black diaphragm on the underside of the primer bulb and don't see any tears in it. I did not replace the fuel filter that sits inside the gas tank. I would very much appreciate any suggestions before I start buying replacement fuel filters, primer bulbs, carb repair kits, etc. Thanks a million in advance! David

I have an engine here. A Tecumseh VH50 as far as I can tell. We're having issues with it. In short, it refuses to start. As far as we can tell we have ignition, it'll turn over, It'll run for about five seconds at maximum if we put gas directly in the ignition chamber to get her started before backfiring. We're not sure what the nature of the issue is. What should we check?