The pumps are new,... not forty years old,... made in Hungary using the original Velosolex machinery if you got it from a knowledgeable parts supplier (like me)...in China otherwise (garbage).
Perhaps the fuel jet is dirty...common problem on a Solex because the passageway through the jet lengthwise is tiny...it's only a 28 cc jet!
Also, think of the Start lever as a choke when first starting the engine, but as a mixture control once the engine is started.
When the engine has warmed up, move the lever left and right to find the best running position.
As long as the lever is somewhere between center and full right, you are fine, no problem.
But, if you have to keep the lever to the left of center for good running, then you have some problem...either too much air in the mixture at the carb (air being sucked into the fuel circuit), or not enough fuel in the mixture at the carb (fuel restriction like a dirty fuel jet or filter).
It is the membrane that does the fuel pump's work. The fuel pump merely contains two check valves.
The crankcase compression and vacuum makes the membrane fluctuate, and suck and push fuel through the pump.
The plastic fuel pump seating piece needs to have a concave side and a flat side. The flat side must face the crankcase. If you cannot easily tell which side is flat and which side is concave, you need to replace the seat, and it must be secured to the crankcase.
Never raise the running engine from the tire.
The clutch will lock up and you will not be able to control the engine speed.
You will not get a true idea of how the engine is running.
And, if you then lower it down onto the tire, you will damage the tire, drive-roller and drive-roller bushings.
DO NOT use any fuel / gas that contains ethanol.
It is damaging to 2-stroke engines in general, and is especially damaging to the plastic and rubber parts of the Solex fuel system.
DO NOT use full synthetic 2-stroke oil.It will leave a film on the sloppy, low-rev Solex engine that will cause over-heating.
Use only semi-synthetic or synthetic blend.
If you e-mail me, I can send you some helpful literature, manuals, guides, etc.