Try avoid judging primer bulbs by how they feel…Sometimes the bulbs get hard when primed, sometimes they don’t. You can’t draw too many conclusions about primer bulb condition by "how it feels".
The easiest thing to do here would be to try a different tank/hose to confirm/rule out these things as problems. If you decide to buy a new hose/primer bulb assembly, I would stick with the expensive OEM stuff. The cheaper stuff won’t last long so there will be no savings.
If the tank/hose get ruled out, then you have one of three possibilities:
air leak in engine fuel lines
restriction in engine fuel lines
problem with fuel pump or pulse line
Generally speaking, an air leak in the engine’s fuel lines should show as a fuel leak when the primer bulb is used. The fuel connection fittings are often to blame, along with old stiff fuel lines, sloppy connections before the fuel pump.
Frank advised removing the fuel filter screen to rule it out as a restriction, have you done that? Look for kinked/crushed fuel lines as well. Did you remove the fuel line from the carb and pump the primer bulb to check fuel flow? The engine’s fuel connection fittings could cause a restriction as well. I suppose the easiest way to rule out some of these issues would be to remove the tank’s connection fitting, and connect the fuel line directly to the inlet side of the fuel pump. I remember you saying you checked the arrow on the primer bulb, but are you sure you didn’t reverse the lines on the fuel pump? Usually, you can’t pump fuel through these pumps in the wrong direction, but I am not familiar with the pump you have.
Are you sure the pulse line is OK? I know you mentioned feeling the pulses in the line, but be sure its connections are tight. I suppose engine damage could affect the pulses through the line, but this seems unlikely if the engine is running OK, but it can’t hurt to check compression. Crankcase leaks will affect pump performance, make sure that fuel pump port cap isn’t leaking.
Once most of this stuff has been ruled out, it is most likely a fuel pump issue, not uncommon with today’s obnoxious fuels…