unfortunately there is no tach on the ski, but how would the rpms be over 1000 if the ski hasn't started yet?
I'd be surprised if the RPM exceeded this threshold value of 1000RPM during cranking, basically just info.
So when the start attempt fails, does the starter motor actually stop, or does it continue running but just stops turning engine (does starter motor become disengaged but still runs)?
I'm wondering if the problem is in the starter, starter solenoid, or start button circuit. Could also be a weak battery/connection or even a possible issue with the lanyard post.
Just throwing out some ideas for the process of elimination.
Physically check to make sure your choke plates are physically closing, if not this could explain a hard cold start. Of course this doesn't have much priority if your starter doesn't crank the motor but mine usually starts within 5 seconds cold with choke pulled, if it's stubborn I'll give the throttle a couple of pulls to take advantage of the accelerator pump feature.
If your fuel pump (on the side of the carburetor) check valves aren't sealing well, the fuel could be running back into the tank, which could leave the carbs dry(almost empty) and require extra cranking time to draw the fuel back up to fill the carbs. You could consider temporarily installing a fuel line from the reserve switch valve to the fuel pump made of clear vinyl. Home depot has this tubing.
The starter issue sounds like something's not working properly though. I think it should crank as long as you hold the start button, even if it takes a long time to pump the fuel back up.
My engine isn't the same model as yours of course, you have the 717.