FWIW, the "primer bulb" is not really a primer at all. It's more of an "air purge" bulb. It does not force fuel into the carb throat like a 'traditional' primer bulb does on a 4 cycle engine. When you push, the air and/or fuel in the bulb gets ejected directly into the fuel tank. As the bulb comes back to its normal shape, it creates a vacuum and sucks fuel from the tank, through the fuel filter stone and into the bulb. A couple of presses and you end up with bulb full of fuel and when the engine vacuum draws in, it's getting fuel instead of air! I've experienced slow return of these bulbs, particularly when it's drawing in 100% fuel.
There is a host of things to look at when troubleshooting a 2 cycle engine. Given your description, I'd be pulling the carb and checking the screen to see if it is clean. You'll know it when you see it. I'd also go ahead and replace the gaskets and diaphragms. Gasket kits are cheap(less than $15) and since you've already pulled it apart, it makes sense to replace. It looks like the carb is a Walbro WT-827. If you haven't tinkered with 2 cycle carbs, go to the Walbro site. They have service manuals for all their 'series' of carbs, this one being the WT series. Parts lists too for each model. They also have some service videos, and good 'ole YouTube has a bunch as well.
Be careful not to blow the carb out with 100psi of air and I'd shy away from shooting carb cleaner through it as the internal check valves can be damaged. The gasket kit is a D20-WAT which is extremely common. The full rebuild kit is a K20-WAT which I would not recommend. These kits are used for many carbs and you will have a bunch of pieces left over as you'll only be using 4. You can pitch the rest!
A final caution if you decide to do this, pay attention as you pull the covers off the carb. On one side, the gasket goes against the carb body, and then the diaphragm(called the metering diaphragm). On the other side, the diaphragm(called the fuel pump diaphragm) goes against the carb body, and then the gasket. Things won't work too well if either of those are reversed!
Good Luck. If getting the carb in shape doesn't solve your problem, we can proceed to the next things to check!