Well, the situation has the same similarities when we talk about treating the water levels as thought they are higher than the basement floor.
You basically answered all your own questions, you are thinking exactly like you should.
Remove the rootcellar creates a hole still into the foundation, the whole area needs to be dug up,just as you said. The waterproof companies will tell you you can just apply some tar coating, not true, that is what is called water dampproofing. You need a waterproof system installed if you need to get rid of the water.
Google waterproof foundations, there are plenty of options.
Hope this helped.