Alternately, if a type time travel is invented where you can change the past, you would expect people to keep meddling with the past until they accidentally changed it so much that time travel had never been invented.

This process would continue, over and over again, and the final results is that the final "stable" timeline will be one where time travel is never invented; not because it's not possible, but simply because every timeline where time travel is invented eventually changes it's own past until it no longer has time travel.