If the insulation resistance is seriously low, such as the mentioned 22kΩ, you can find the problem using either an insulation tester or multimeter, as both will respond to that resistance. What you cannot do with the multimeter alone, for the reason that Radiohead clearly explains, is to subject good insulation to a stress test to prove that it is really good.
As an analogy, if water is gushing out of a faulty pipe joint, it doesn't take any special pressure test equipment to discover where the leak is. But, if the pipe appears watertight when there's some water standing in it, to say with certainty it won't spring a leak in use, you've got to put on as much pressure (or more) as it will have to withstand in use, and that is the specific function of an insulation tester.