Dead of winter and dead centre appear to have different origins, and their relation to being in the middle seems to be slightly coincidental.
the dead of winter the coldest part of winter.
The sense of dead here and in the previous idiom developed in the 16th century from dead time of —, meaning the period most characterized by lack of signs of life or activity.
The earliest recorded use of dead of night, for "darkest time of night," was in Edward Hall's Chronicle of 1548: "In the dead of the night ... he broke up his camp and fled." Dead of winter, for the coldest part of winter, dates from the early 1600s.
As I understand it, the coldest part of winter is after the middle of winter.
Meanwhile, Dead Centre relates to engineering, somewhere in the 1800s, and sources suggest its related to either top and bottom of a crank rotation where it cannot be turned by the con-rod, or the very centre of something being turned on a lathe, a point with minimal movement.
Strange question.Not being pedantic but merely wanting to clarify is it Dead CENTRE or Dead CENTER?