Does it make sense to "pause" an academic career so that I can pursue financial independence and self-fund my research?
It seems implausible that your plan can work: You can't hope to draw more than 5% off an investment per year, so even a moderate income of $50k (or equivalent in your country) will require you to accumulate about a million dollars. It seems improbable that you will be able to sock away this much money in 5-6 years, post-taxes, unless you have a pre-tax salary on the order of half a million dollars.
So what that means for your current situation is then that you'll have to save for far more than just 5-6 years -- and that you'll miss some of the best years of your life where you can establish a career, a family. In other words, I think that the premise in the title of your question is flawed and that you'll miss out on some of the best parts of your life.
(As a side note: In your 30s, you probably will want to have a partner and family at some point. It's conceivable that you could live of less than $50k per year if you're living by yourself frugally. But you can't do that if you have a family and children: They have needs that may exceed your own, and what's acceptable as a living abode for you may no longer be for your family.)