This is good, standard advice (if you can play it slowly...)
Once this is done and the notes are under your fingers, though, most pianists would probably say you have about 80% of the work on a piece left. The next most important thing next is to train your ears to get yourself to the next level. I highly recommend recording yourself in small ~30 second chunks and identifying what you want to change. At this point most progress will be made from listening and analysis, not from fingers on keys.
Of course, this is hard work and requires proper motivation and training, for which a good teacher has no substitute.