The glaring philosophical issues of time travel
The thing that bothers me most about movie and TV time travel -- far more than causality paradoxes, which will probably sort themselves out in the end -- is that the idea of "changing the timeline" raises profound philosophical questions that almost never get addressed. For example, if lots of people die, then the hero goes back in time and saves them, so now they are not dead and never have been... what does that mean? Did they experience "being dead" and then un-experience it? If you're of a religious bent, does that mean their souls were in Heaven but then got yanked back, leaving God going "hey, where'd they go? they were here a moment ago!". Even if you're not, what does it mean for the nature of experience that something can become un-experienced?
Travel into the future raises even more profound questions about the nature of consciousness and free will. For example, suppose I travel to next week and meet my up with my friend Alice (let's call her Alice+7). She appears to be -- and believes herself to be -- a perfectly normal, conscious, freely-acting person. And if I ask Alice+7 for her personal perspective, she will probably say that she exists and is conscious at that instant in time, continuously moving forward into the future. But if I travel back to my own time, Alice+0 will say the same; and so would Alice+14 (Alice two weeks from now). It seems like there is not one Alice existing from moment to moment, as we normally think of our conscious selves, but an infinity of Alices each existing in their own moment, any of which I can visit, and each convinced of their own continuity (and of course, if Alice had the time machine she would say the same about me). If the future (relative to my personal present) already "exists" in some sense that allows me to visit it, this is the inevitable conclusion.
Most TV and movies ignore this completely -- in fact, they act as if the perspective of the protagonist is uniquely special, that his or her "now" is the one definitive, privileged, real "now".
And yet... when a couple of Star Trek characters travel in time to save the day, it's as if they say "well, we just had an adventure that not only raises profound questions about causality and paradox, but also throws into doubt all our concepts of self, consciousness, free will, and the nature of experience. Let's never speak of this again."