You're acting like a child who keeps asking "why?" to every statement. Don't stay hung up on "is my idea of reincarnation real, or not?" - because that is just a concept. What's the difference between two trees? What did you bring with you when you were born? Whose mind would be reincarnated? There is no mind - the concept of a mind exists; and there are brains. Science is a concept. Religion is a concept. I never equated them, because what I am describing is not a concept: reality itself is not a concept. Your either/or viewpoint will get you nowhere in these matters: there is no alternative - it is both, and neither; non-dual. I'm telling you that the starting point for this experiment is to look at reality closely, without trying to complicate it by saying it must be this and not be that. It is what it is.
Of course, this makes no sense, because I can't make that step for you. It could take minutes, or years, to get the point. Be instead like the child who sees the order in clouds without labeling them as seeming to be this or that as they move across the sky. The experiment is experiential to the individual; do these things, and expect this result to inevitably occur. Do not these things, and it will not occur. Thousands of years of the same results, and yet you ask for evidence to be produced by others instead of yourself.
Don't misconstrue this as a personal failing - everyone starts from the same confusing position you are expressing. That each individual must overcome this futile attempt at reasoning and thinking their way to liberation is only a problem when you are determined to solve the problem in that way.
Look. Spin a pen or shoot a basket until it can be done without effort or thought - who is it that has learned this? Motor memory, brains dissected and labeled; etc., but all that is useless towards learning or doing. Would it ever be the same to describe how these actions are done with words as it is to do them without words or thoughts? I can spin a pen skillfully for a long time with the simple directive of "don't drop it", and my fingers do it for me. Over time, they avoid actions that did not lead to the desired outcome, moment to moment. When that brain that has learned that action has been dissolved, who would expect it to be innate in a new being? Yet the knowledge of how to do this thing was never mine; it is available to all to use.