The jet pump assembly is worn out as you have discovered. Jet pumps are very sensitive to any wear or air infiltration which causes cavitation and loss of thrust. When you get the new pump look up the impeller to wear ring clearance specs and measure it with feeler gauges. If it looks like the wear ring has some decent wear or if the impeller has gouges or bad dings on the leading edges I would replace them. Also there should be gear oil or some kind of lubricant that was inside of that cone on the pump to protect the bearings. Hope this helps.
I just converted my jet ski to premix oil and bypassed the oil pump, this is the number one cause of engine failures because either the pump goes bad or the tubing gets brittle and the engine doesn't get the lubrication and then "no bueno" go premix is the way to go.
I have the same JetSki. I was able to get it started things to your video but now is pushing out a lot of white smoke from the exhaust with some oil and water. Do you have any tips to stop the white smoke and the oil coming out of the exhaust