No it doesn't reduce it, that would be straight up magic. If the key is in the interval [1, L] and you shift it to [-L/2, L/2], you can never know if the shifted key to look for is positive or negative, so there is no new knowledge to take advantage of. It would only reduce to half a brute-force search, but brute-force is not optimal at all for this case, so it's never used. What you get instead, is an interval of the same size, and the knowledge that once you find a target point, it can either be the one on the left or the right, relative to the generator point (middle point). That means matching the X coordinate.

Using the symmetry is pretty much the basis of the optimized BSGS and 3/4-kangaroo algorithms, to go below the 2*sqrt(L) number of operations (required in the general case of a group without symmetry).