I am going with the rectifier on this one. When the rectifier starts to crap out many times the ski will do intermittent funky things like what you have described. When the rectifier is not working properly it will also not charge the battery, which might be why you cant get it to turn over.
What condition is the battery in? Charge it fully and see if it will turn over normally.
Now to address the intermittent running issue,
To start pull the red wire off the rectifier in the front ebox, that will take it out of the equation. The ski will still run fine but it wont charge the battery.
After that make sure your battery is fully charged and run it with the red wire off and see what that does. If it runs good with the rectifier unplugged, then you need to replace the rectifier.