Can a 3 years PhD make an independent researcher ? Or how many post-docs after PhD to be an independent researcher?
As a PhD student, I think you are doing an original research, so should be developing a new method, model, formulation, etc so that you can solve many other problems or cases by means of your own formulation. Even, by your model, you should be doing some verifications and find results others found in their publications. So, during your PhD you are being prepared to do independent research.

I think becoming independent in every phase of life should be assessed based on the person's abilities and the outcome of his activities.

If the person can find/understand the things he has to do without help of others, consult his problems with others, does what he has to do to reach his goals and finally finish his jobs/projects and has an income; then he can be considered an independent person.

In academic world, the case does not change so much. If the student can review literature, read books and papers and understand them, find a new problem to solve, start to work on the problem and consult his problems with another person; and finally submit and publish his paper in a journal; then he should be considered an independent researcher.

Despite the thing in your question that you mentioned how many post-docs is needed to become independent researcher?; I don't believe a person need many post-docs. Even, some researchers do not have any PhDs and they do perfect researches. The important part of doing independent research is the art of problem solving. And this may be gained by no PhDs or by hundreds of post-docs! So, I encourage you to successfully finish your PhD, and try to learn how to solve your academic problems and how to get them published. After you graduate, you will be able to do research by your own too.