Spending time doing independent learning, even if you have given yourself some structure, misses some of the important things you get in a PhD program. Your advisor guides you, but they also challenge you. They hold you accountable. Some people may be able to challenge themselves, but most people don't have the necessary experience to know the difference between a "cool hack" and novel research. Your advisor has more experience and can tell you what is worth pursuing. Plus, to complete a PhD you have to do all the requirements- even the stuff you don't like. A self-structured independent study program is probably going to be light on the stuff that you don't find fun.
There are a lot of downsides to doing a PhD, and I'm not saying that it's for everyone. Independent study is fine. But I'm going to look skeptically at anyone who thinks their own independent study program is the equivalent of a PhD. I would probably be less skeptical if their independent study resulted in publishing a few peer-reviewed journal articles. The title of PhD doesn't just signal that you are capable of learning, it's also a signal that you can work within the current research environment, have your work accepted by other experts in the field, and in general just do a bunch of "intangibles" that aren't signaled by you working on your own.