Will it improve my PhD applications if I take a gap year to work on research?
Your situation sounds similar to mine. After finishing my undergrad and not getting a PhD position, and since it was too late to apply for a full MSc, I moved to another country, enrolled just in the thesis and I worked on it for a year. After that, I got a lot of experience, a good recommendation, and a position. Now, I was lucky I could move and found a university that the only requirement for registering was finding a willing supervisor. So, I won't say necessarily that everything is lost, but it may be difficult depending on your situation.

I would advise against being fully independent researcher. You are fresh from undergrad, and with not so much experience in the field. If you work without adequate supervision chances are you will be reinventing the wheel, walking down paths that lead nowhere, or creating a beautiful yet useless theory. Your priority should be finding a professor that can guide you and provide some interesting problems to work on and feedback.

You should also keep an eye to applications in Europe: positions are opening and rolling all year round.