I would wager that neither of those methods would actually work. We know that quirrel, snape, and other very intelligent wizards know both about the muggle world and about time turners. An idea about Arbitrage harry had within a few days of finding out about the ENTIRE wizarding world is extremely unlikely to have never been thought of. Harry is able to come up with new techniques and things no wizard would come up with but things like partial transfiguration took him days and are extremely non-obvious. Plans to make millions upon millions of dollars are something people have all the time.

Another important concern: Be careful what you call a "trifling" amount of money. You can get a LOT MORE done with a hundred thousand dollars than you can with zero dollars. You can hire bodyguards, you can rent a house, you can buy equipment, there are a vast world of things you can do that are prohibitively expensive without needing to be bill gates. We talk about vast amounts a lot when we're talking effective altruism and existential risk ending but Harry's not fighting to end hunger, he's simply fighting a relatively small-scale war.