Most larger engines (>3.5 HP) use a tank with gravity feed, or a fuel pump mounted on the side of the engine to pull the fuel out of the tank, and the carburetor has a float that operates a needle valve to maintain fuel level in the 'bowl'. In fact, I haven't seen a modern bowl mounted carb in a while. WIth gravity fed systems, there is usually no primer bulb because the mass and volume of fuel in the tank is sufficient to easily feed the carb [especially true on horizontal shaft engines], normally mounted BELOW the tank, but that isn't always the case. An external fuel pump is normally mounted somewhere on the same side with the 'breather', which is the crankcase vent that supplies pulsed air to move the diaphragm and pump fuel to the carburetor, much like you're doing when you push the bulb on one that has a tank mounted carb. Some may have a line that goes to a port on the carb that uses intake vacuum to operate. The external pumps move a lot more fuel than the much smaller internal pumps on smaller carbs.
As
@ILENGINE pointed out, if the carb is mounted on the tank, the fuel tank is usually the bowl of the carburetor on the smaller engines, but he skipped a few steps as it seems you're not getting a constant supply to the carb, never mind getting it into the venturi. A tank mounted carb gets pulses to operate the fuel pump (small diaphragm in the carb) from the intake stroke of the engine oscillating the vacuum level in the intake. When you remove the carb, you'll see a small port that runs alongside the main intake port, usually made into a piece of plastic, that connects to the galley in the carburetor to operate the internal pump. That pulsing causes the diaphragm to move, liquid on one side, air on the other, and push fuel to the carb reservoir. This allows the tank to be mounted below the carb, making a more compact design. The rubber or neoprene bulb is just a way to get fuel there quicker bypassing the diaphragm in the carb, and alleviating a LOT of pulling on the cord to start the engine when it's cold and hasn't been run in a while by forcibly pumping fuel through the delivery circuits. Once primed, liquid is easier to keep moving than it is to get it started moving, (because air compresses but liquid does not) which is where the diaphragm fuel pump in the carb comes in. If it isn't priming when you press the bulb, it sounds like the reed valve gasket on the carb has something stuck in it and won't let one of the valves close, or is possibly installed backward so that the flaps on the valve do not align properly with the ports to the tank and bulb. One valve allows fuel to be pushed back to the tank, the other allows fuel to be sucked into the bulb. The reed valve looks a lot like a thin rubber or maybe mylar gasket, usually between the carb and the tank. Once the engine starts, the pulsing vacuum between the carb and the cylinder moves the diaphragm and makes it do the same thing you're doing with the bulb, using the exact same reed valves. On larger engines with external fuel pumps, the fuel tank can be mounted most anywhere because the external pumps have considerable lift and pressure. They move a LOT of fuel, as needed, or they simply keep the supply line to the carb pressurized, relieving back to tank side line if the fuel isn't getting used.
How much fuel is actually used by the engine will depend on the orifice (jet) size and air flow. A plugged jet may give the same no-start condition, but that completely different from fuel delivery issue. This usually doesn't result in no-start but poor fuel mix will likely not let the engine continue to run. There are normally two jets, a low speed jet for idle, and a high speed jet for (you guessed it) high speed. The high speed jet is normally larger and has a more direct path to the fuel source for better delivery, and usually between the throttle venturi butterfly and the intake port. When the butterfly opens and more airflow goes across the top of that jet, it creates a pitot suction in the tube to pull fuel from the carb reservoir. The idle jet is usually before the butterfly, and near a hole in the throttle butterfly to provide just enough fuel to idle, however, without that additional fuel, even the high speed mix will be wrong, and the engine will run poorly. The low speed jet is normally what gets stopped up mainly because it's smaller (somewhere in the .003 and .010 range, perhaps even smaller), and won't let the engine idle or at best it will run poorly above idle.
SOME carbs are now completely sealed, and unserviceable, and meddling with them to try to relieve an obstruction normally results in complete destruction. Compressed air is OK to clean out the galleys and maybe the jets (normally take a little more persistence), but don't use it unless the carb is completely disassembled and the gaskets/pump removed. You will destroy the valves and/or diaphragm pump. The idle and high speed jets were at one time both adjustable so you could tune the fuel consumption and performance. First came locks on the adjustable jets to limit adjustment range, then came non-adjustable jets. If you get familiar with carburetors, they're actually easy to work on with the right set of tools. A little better performance can be had by tuning one rich, but it isn't real good for the environment, nor is it good for the engine.
Notwithstanding, some gravity fed (tank mounted above carb) engines don't use a separate external pump, yet still have a carb with a built-in pump, which also includes the bulb for priming, because it's harder to raise gasoline high enough above the carb to get reliable flow with low vacuum in the intake to operate the pump and also keep the engine compact enough for walk-behind equipment. If the tank is mounted high on the side of the engine with a short line to the carb, it's VERY possible that it still has a bulb for priming, especially on newer engines. The carb on that engine still works similarly, but relies on fuel to flow to the bowl from the tank via gravity, then the bulb sucks fuel from the bowl to purge air from the diaphragm pump, and the supply for the jets is the bowl on the bottom of the carb. If there's air in the pump, the pump doesn't move the fuel. Some of it will be shot into the venturi to provide a rich mixture to help the engine start with reduced compression, but most is returned to where it was pumped from. It's a lot harder to flood a small engine than it used to be.
This whole process relies on fuel not being compressible in order to move it. If there's a bowl on the bottom of the carb, it's completely possible that the needle valve that normally controls the bowl fuel level is stuck, unless the 'bowl' is the tank. No float valve in that, just one short and one long pickup tube. A plugged/stuck float valve will create exactly the same symptoms you described. Most float needles have a neoprene tip, and do not like ethanol. Most gasoline these days has SOME ethanol content. You say you replaced the carb, so I wouldn't think that's the case, BUT, stranger things have happened with new carbs. The float may move, but that doesn't mean the needle is. That's eyes on information, or you can put a little fuel in the line and see if it comes through when the float is down, and does not when the float is up. (This requires bowl off, btw. Not the bulb, the bowl).
Carb work is very detailed and meticulous. I find it actually challenging and fun. If you'd post a picture of your carb and engine, you'd make it a lot easier to help diagnose your issue. If you have replaced the carb, is it possible you left a gasket off or put it in upside down? Is it possible your original problem is still there? (trash in the fuel circuit). When doing carb work, the smallest particles matter. Dust will clog the jets and galleys in a carb very easily. Keep everything as clean as possible when you're working on the carb. They're very sensitive to debris. Keep a can of spray carb cleaner handy while you have it disassembled. DON'T use it to clean the carb on the engine, particularly not on the inside, unless the engine is running. You're not likely to clean out jet obstructions with carb cleaner. That normally requires disassembling and tiny jet files or drill bits to clean them out. If you're not comfortable doing that, best let someone else who is.
