If the Earth stayed fixed in space on one side of the sun, yes, noon at one point on the surface would be midnight six months later.
But the Earth also revolves around the sun, which exactly compensates for the Earth being on the opposite side of the sun once per year. So the six months net change in direction from a four minute shorter revolution (than the length of the “day”) means the same side of the Earth points towards the sun at noon, six months later.
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