The time travel bottleneck applies to all civilizations. They achieve a certain level of physical knowledge and develop the ability to travel back in time. They develop this before intergalactic travel, and this has the potential to create time wars on their planet.
This level of physical knowledge is relatively easy to achieve once a civilization achieves a certain level of physical mastery, so there is a limited time for a civilization to develop before rogue groups without regard for the safety of others develop the ability to travel time.
Any first time civilization that develops time travel will probably have emerged from great wars and suffering, but understand that altering events may not lead to their development of time travel. They would know that the development of time travel might be achieved by others who may alter the timeline at various points, so the time race begins.
Perhaps they discover that they only develop branches in the multiverse, and must develop ways to travel between the branches of the multiverse- if this happens then there is no danger to any timeline, they just develop additional ones. In the multiverse scenario, it would be good to send someone with all of your technology back to a point in time that they could establish a way of bringing peace to enemies, so the multiverse is not littered with branches with many vengeful hate filled beings.
Civilizations that prepared for the knowledge of time travel would need to have a plan in place to go back in time and obfuscate the knowledge that leads to technology that leads to time travel. They would have to corner the discipline of physics in order to prevent anyone else from developing time travel, or else at any point in the timeline in which beings who are not part of the time lords begin developing physical theories that lead towards time travel, they would have to destroy those beings or limit their ability to advance in the direction of knowledge.
After several cycles, in which technology is brought to the past and developed further, the time travelers would hopefully develop technology to limit the development of time travel technology by other groups. They would want to increase the knowledge of other technology, but not lead to time travel tech, until they reached a point that people would never want to travel back in time to alter things, by instilling in people great value for beings that they grew up with, and the knowledge that time travel could destroy everything that everyone had worked for.
People must have a built in switch that does not allow them to time travel, so they do not ruin things, a switch that is only turned off if they have knowledge that they should do so for the good of the whole civilization.
I think I need to start watching Dr. Who again.