I am not familiar with the suzuki at all. All I can offer is Yamaha experience, and I can't imagine it being much different.
Anyway, the prime starter unit has a plunger on the end of it. It retracts into it's housing when cold, and extends when hot. When it's retracted, it opens up a fuel and air passage. When it extends, it closes off the passages as the wax inside of the prime starter warms up. It doesn't click in and out, it is slow and usually works off of alternator voltage. Higher voltage from a faster spinning flywheel=faster closing prime starter. Anyway, that passage leads into the carb bowl, through the gasket usually, and down inside the bowl usually lives a jet-or at least an orifice. Something to control the volume of fuel passing through it, and that could be where the problem lies. Pull the carb apart and take a good look at how everything works and I bet you'll find the problem. I've run into some Yamaha's that had 4 carbs and 2 prime starters, and some 3 cylinder engines with 3 carbs and only one prime starter, and twin cylinder motors with a single carb and one prime starter unit. They're all different it seems like. The F50 carb'd motor, for example (as well as the very early 4 cylinder F40) both have 4 carbs and only 2 prime starters. The primer passage had a nipple on it which leads to the other 2 carbs that do not have the primers. Also, some of those I've had experience with also have 4 different main jet sizes, 4 different pilot jet sizes, etc. I had one at one point (and absolutely loved it) and it took me a couple years of fiddling with it to find out that the jetting was staggered and that the prime start bowls (and gaskets) were DIFFERENT than the non prime start bowls. Pretty complicated setup but worth mentioning in a case like this. Some of the 2 stroke 40's also had prime start, 3 cylinders, one prime starter. Bowls look identical and they'll fit on either carb body. But there is a difference internally and when pulling one of those apart, the bowls need to be marked, etched, stamped, whatever. Then there was yet another 40 that had a remote prime starter...which bolted to the side of the center carb, and had teeny tiny passages inside of both the bowl, the carb body, and the primer assembly....if one of them was severely plugged, the entire carb had to be replaced (which ain't cheap).
Best bet is to pull the carbs apart, find the one that has a prime starter (it usually looks like a solenoid sticking out of the carb, with a wire or two on it), pull it apart and then have a look-see how it works. Bet you'll find your problem once you figure how how everything functions.
I sure wish that they all had choke plates like the days of old, but thanks to the good ol' US EPA, that ain't gonna happen anytime soon.
