I was out on the water last week and I ran into a similar issue when I was just tooling slowly around the shore looking at property for sale. I sucked in a weed and didn't realize it until I decided to head back to the boat launch. Same kind of thing, it would rev and shake like mad but not go anywhere. I got underneath it and dislodged the weed (it didn't take much of a weed to clog the intake) and it was fine.
There is no water in the pump when it's on the hose, so you won't create that low pressure in the pump intake that boils the water and creates the cavitation that leads to the vibration. When you are in the water, then the flow restriction creates the low pressure on the pump inlet that leads to the cavitation that you feel as a vibration and you don't go anywhere.
I'm new to jet skis, so I'm not any kind of expert on these things yet, but it sounds like something is restricting water flow to the pump. Maybe get a better look inside the grate? I'm sure one of the forum experts will give you better advice than I just did, but that is my guess based on my experience with a weed clogging my intake.