I am expanding my comments into an answer as requested by jakebeal.
It is a basic tenet of academia that the identity, geographical location, or institutional affiliation (or lack thereof) of a researcher should not matter when it comes to publishing their results; all that matters is that the results are of good quality and of interest to the research community. According to this ideal, academic research is an enterprise open to anyone, anywhere. So, yes, your goal of performing independent research is a noble one and certainly has value, both to the community and to your own professional prospects. However, it is important to stress that this value will be created primarily if and when you are able to create research that's good enough to be publishable in a peer-reviewed journal, and not before. If you are serious about being an independent researcher (or any kind of researcher, really), publishing your results should therefore be your goal.
In the comments you point out your belief that journals rarely publish the writings of independent researchers. I agree that that is very probably true, but one should be careful not to draw the wrong conclusion here. It is not so much that journals are snobbish and elitist and simply refuse to seriously consider papers from an author with no institutional affiliation (although, to be sure, one cannot entirely discount a small and mostly unintended psychological bias making it marginally more difficult for an independent researcher to get their work accepted). Rather, the main reason is that independent researchers face multiple obstacles to creating good work in the first place. Those obstacles include:
Independent researchers don't have a lot of contact with other researchers from the community who can tell them about the state of their field and give them ideas for good research problems to work on.
Independent researchers usually don't have anyone to criticize their ideas and give them feedback. Such feedback can be quite crucial to making progress in one's research.
Due to their lack of contact with their discipline, independent researchers often don't have a chance to become well-versed in the terminology of their field and in the cultural norms of how research in their field is typically presented. They may develop good research ideas but do so in an insular fashion using their own private language, terminology and presentation style, which makes it much harder for them to communicate their ideas to others.
Importantly, independent researchers are not paid by anyone to do research, which usually means that they have to spend most of their time and creative energy working a "normal" job. This puts them at a significant competitive disadvantage compared to non-independent researchers.
Let me now get to your question. Even if you never publish your results, I still think there is value for you to try to work on some research problems in your spare time. Aside from the fact that research can be a lot of fun, it will certainly develop your intellect and probably help you develop professional skills that will come in handy at one point or another, whether you continue to pursue a doctorate or end up in an industry job. However, as far as things that you can put on your resume are concerned, I don't think private research writings that you wrote for your own amusement but are not of good enough quality to be published in a journal, or at least as a conference poster or on a reasonably good quality online blog, will be of much value on a resume.