In what way doesn't warp drive make sense, scientifically? It's a completely made-up technology.
Special relativity, causality, and FTL: you can only pick two, all three are mutually contradictory. It's not a tech barrier, it's a physics barrier; you can't tech your way out of literally any usage of FTL resulting in time travel under special relativity as it is in our universe, because the order of events is only preserved for space-like separated events. (That is, for two events that are far enough apart in space and time that light couldn't pass from one to the other, the order in which they happen depends on your frame of reference. Not just the order in which they appear to happen, it's not a travel-time-of-light issue, but the order in which they actually happen.)
There's more information on it here, and a good illustration of it in the Ladder paradox here. But for a quick illustration of the ladder paradox, consider this: you have a barn with a door on either side, and a rod that's too long to fit inside the barn. The rod moves towards the barn fast enough that its length is shortened in your frame of reference small enough to fit within the barn, and you have a timer set up to close both barn doors at precisely the same time when the rod is within and open them again before the rod reaches the far side. What happens from the rod's frame of reference? From its frame of reference, it's too long to fit within the barn, and in fact the barn itself would be length contracted as well to be even smaller, so how can both doors close at precisely the same time from the frame of reference of the rod?
The resolution to the paradox according to special relativity if you run through the math is that in the rod's frame of reference, they don't close at precisely the same time; the actual time of the events of the two door closures are different from the perspective of the rod, purely because the rod is moving really fast. The door ahead of it closes and opens when the front of the rod enters the barn, and the door behind it closes and opens when the back of the rod enters the barn. And there exists yet another frame of reference in which the door behind the rod closes and opens before the door ahead of the rod; again, a frame of reference in which it actually does, not just one in which it appears to because of the travel time of light.
Because of stuff like this, literally any FTL drive would be violating causality for some pair of events every time it was used; you literally couldn't use it without having something happen "out of order".
Space and time are much weirder under relativity than even things like time dilation and Lorentz contraction suggest, and it's far more complicated than just the speed of light being a barrier. There are a ton of other variations on the ladder paradox in that Wikipedia article, there's a lot of alternate formulations that are also really weird from our usual perception of space and time; it's definitely worth a read.
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