Does the latest Minute Physics video on time travel in fiction strike anyone else as not being very much about physics?

You can watch it at https://www.youtube.com/watch?v=d3zTfXvYZ9s. I'm not looking to start a hate thread, just want to hear some opinions.

Basically, the video analyzes the depiction of time travel in many works of fiction, looking at what happens when the universe splits into different timelines and stuff. The conclusion seems to be that (out of the examples presented) the only realistic "time travel" happens in Ender's Game, which is really just time dilation, and furthermore that the only actual time travel which makes logical sense is the one in Harry Potter. I agree with both of those things.

Still, there's a feeling I always get when people ask about time travel in a physics context, which is that it doesn't really belong there. Not only is time travel not possible as far as we know (which is not a big deal in sci-fi), almost every description of it seems to rely heavily on the concept of a person with consciousness, and what happens to that consciousness and free will when the person travels in time (take Groundhog Day for example, where the thing doing the time travel is Bill Murray's mind instead of any actual physical object). As far as I know, these are not concepts that physics can really talk about, being more in the domain of psychology, neuroscience and even philosophy.

Do you think the subject of time travel as usually depicted in fiction (with people traveling in time, timelines splitting and all that good stuff) is something physics can talk about? Or is it beyond what physics covers? Or just logically inconsistent?