I don't know that this technique is a big win for this kind of memorization. For example, multiplication of digits probably needs to be in L1 cache. You don't want to have to search for the result in your memory palace, which seems more like main memory.

Indeed! There are some things that you need to force feed, and others that you can place in a memory palace or use in an associative array of thoughts. I.e. multiplication tables or foreign basic words need to be in your L1 or L2, but the causes of the First World War (unless it is something you are deeply related to) can perfectly be outside it.

To continue with your example, multiplication of digits gets to be in L1 having lived in main memory for long enough, and being retrieved from main memory often enough, to stay there. The same will be true of anything which is accessed that frequently, but unless things are in the main memory to begin with, they can't possibly get promoted.