I can’t see any downside to adding a primer bulb. There is no way that forcing the motor to build fuel pressure while it is cranking is going to be helping with this cold starting issue.
Given your logic of “I can’t see any downside to adding a primer bulb.” would that mean you would add a primer bulb to the EFI fuel system on your auto, truck or ATV?
As far as your second statement;
In the case of a EFI fuel system, the “motor” has nothing to do with the fuel pump pressure. The fuel pump is electric and is independent of the motor. Because the fuel pump (both the lower pressure pump and the high pressure pump) get their power from the battery, a low pressure condition may be an electrical issue.
The available voltage drops when the battery is called upon to crank the engine. Therefore there is less electrical power going to the fuel pumps, which may in turn produce less fuel pressure. This is one reason motors now require higher CCA than was called for in the past.
A primer bulb, will not increase the output of high pressure fuel to the injectors.
If the “cold start” issue is too low of pressure delivery to the fuel injectors, it could be one or more of several issues within the fuel system or electrical system.
On a personal note; The EFI fuel system on my boat does have a primer bulb. I can’t remember the last time I ever even touched it. Maybe that is why some systems don’t have one, the engineers have eliminated another potential failure point, by eliminating a primer bulb which no longer is needed in their system.