We've probably talked about this before, but I think the intuition about how time seems to revolve around the moving present differs from the scientific understanding of time. Science posits whatever it needs to to explain the data, and those posits depend also on what the math allows.
So we have these three things, according to our concepts of time and causality: past, present, and future. Are they all equally real? Our experience suggests that only the present is real since that's all we're directly aware of. But we explain the causal patterns in our experience by positing the past and the future (and cause and effect). So events seem to be moving from the past to the future, in that the past comes first and produces the future.
Following Einstein, though, physicists say that our perception of time is illusory since it's based on the limitations of consciousness. There is no moving present, and the past and future exist as a fourth dimension of the block of space-time. The future and the past may equally exist, but we're not aware of those parts of the "fabric."