The problem is all objects need to have a top speed. If they don't, you run into all sorts of trouble - from gameplay issues (imagine how it would work/look if a missile or ship could travel across several screens in a second!) to engine issues (you could miss collisions if objects travel very far in a single frame, and would need to use a different, much more complex algorithm for collision detection).
Missiles *do* retain the firing ship's velocity, though. This is most apparent in the case of bombs, which rely on the velocity of the launching ship to carry them to the target.
Might be fun sometime to play with a missile that had some low-ish acceleration and an absurdly high max speed, though.
Seems like it, doesn't it? From everything I've tried that was like that, though, it's actually just frustrating to watch because it feels so sluggish.
That was the original idea for Pilums. What'd happen is they'd always miss their target on the first pass - can't accelerate fast enough to track it, and the high speed is a handicap in this case, since they have less time to adjust for target course changes. Then, they'd literally take about a minute to turn around and reverse their velocity, only to miss on the second pass or flame out.
I suppose it's probably possible to work it out to where it'd feel better, with some combination of speed/acceleration and AI behavior, though.