I think part of the problem when pulling off time travel in any media is the scope of said media.
Chrono Trigger has a relatively small world by RPG standards (the different time periods make it seem bigger than it actually is physical map-wise), which allowed the game makers to allow players to travel through time at will. The restrictions on time periods one could visit helped keep things manageable here obviously.
A lot of the other games that have time travel tend to only allow players to time travel in a very restricted way (ie Tales of Phantasia does not allow the player to move at will between time periods), which mostly prevents plot/logic holes for the most part.
I think part of the reason Dr. Who time travel works is that vagueness everyone mentions. There is established lore (just check the wiki pages), but it tends to leave a lot to the imagination--especially the newer iterations have generally glossed over a lot of the details in past series (see: a metric ton about Time Lords and Gallifrey, including the regeneration limit, which were literally Deus ex Machina'ed away to simplify things). Sure, this stuff gets referenced sometimes, but the newer series mostly started over and focuses more on adventure elements rather than a lot of the detailed world-building that went on in the original (IMO not a bad thing, just different).
So, I guess my two rules for good time travel in media are as follows:
1. If you plan on doing complex world/lore building and incorporate time travel, you should either A. restrict the scope of the world you build or B. restrict the role time travel plays.
2. If you want to time travel to be a central part of your narrative, you should reduce your world/lore-building to reduce the chances of detail problems arising.