How to approach a professor for an independent study course?
Yes, at least in mathematics, beginning grad students often cannot make informed choices about worthwhile directions of study, and there can be advantages to getting more-expert opinion. This is not at all to say that you should not follow your own curiosity and intellectual impulses! But only that you should also be open to expert advice. So you might best view arranging your independent/directed study as an enterprise whose goals can be negotiated, hopefully with all parties thinking of your best (long-term?) interests.