Why is it half and not second?

I suppose because, in the everyday speech of ordinary people, the use of half long predated the other fractions.

This is the earliest entry in the OED, from the year 835.

Charter in Old Eng. Texts 447, & him man selle an half swulung an ciollan dene.

Whereas the earliest reference to third as a fraction (as opposed to an ordinal number) is from half a millennium later ▸c1384.

Bible (Wycliffite, E.V.) (Douce 369(2)) (1850) 1 Macc. x. 29 Nowe Y assoile ȝou..of tributis, and I forȝeue to ȝou the pricis of salt, and forȝeue crownys, and the thriddis [a1425 L.V. thridde part] of seed.