Carbs!
Buck, pulling on the choke cable, amounts to changing the air/fuel ratio, in this case, to suck more fuel into the engine so it will run. When you first start an engine and it's cold, the choke helps to run the engine richer until it heats up, then you don't need the choke.
It sounds like to me, that your main jet could be plugged on either one or both of your carbs. As old as this ski is, I wonder when the last time they were rebuilt. You may also have a bad diaphragm, which when you push down on your throttle, causes the pilot jets to give the engine a burst of fuel for take off.
I didn't read far enough into this chapter to see if your using premix or have the oil injection tank. If your plugs are showing white ash, like your running really lean, I'd also want to check the oil system outside of the carbs. If your ski is using the oil injection system, from others I've heard talking about this, you might want to start premix (40 or 50:1)..........if you already use premix, I'd look at rebuilding the carbs. Also, check the syncro bar to make sure your carbs are in sync. If you haven't already done it, become a premium member and you'll have all the access you'll need to confidently rebuild the carbs......and if you get stuck, hey, we're always here to help you out!
BTW, I don't think you've got bad crank seals. Just doesn't sound logical. If it were a certified mechanic telling you this, then maybe it's possible but to me, it just doesn't sound accurate.