There can be stable timelines with time travel - there's actually 3 states:

• Perpetual instability, where the timeline changes each time the time machine is used but never reaches the same state twice

• Perpetual cyclic stability, where people's actions in modifying the timeline lead to it eventually reaching the same state, eg. you go back in time to kill someone who becomes evil and oppresses you but the near death experience leads them capture you, so you can't time travel any more, and to blame your people and start oppressing them, leading to the same actions

• Stability without time travel, which is the default state but incredibly hard to get once time travel is invented as with nobody to stop time travel being invented it would probably get invented again, however parts of a cyclically stable timeline could have nobody having access to time travel, but any actions by time travellers to stop time travel would likely lead to the second rather than third option