turn it between centres
It is a twist drill. How would you put a centre into the pointy (cutting) end of it? What tool would you use to drill a centre hole in a tool that is itself designed to cut metal?
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It the shank really needed turning down, bore a 7/8″ hole into a piece of something and use body filler to glue the cutting end of the drill bit into that. Once the filler sets, grip that item in a headstock chuck, Use a tailstock centre at the other (square end). If the square end does not have a centre in it, it should not be hardened so one can be formed.
The drill bits are not used in a carpenter’s brace – they are used in a ratchet drill. In a YouTube video by DoubleBoost, I have seen an adaptor that goes from female square tapered to male tapered Morse taper (I think it was 3 Morse).
There is extensive discussion and an original catalogue picture here:
https://www.mig-welding.co.uk/forum/threads/drill-bits-with-square-taper-shank.73650/