Interesting further replies. I perhaps didn't make it clear in my original post that the 'self feed' was severe enough to pull the taper from the tailstock bore, so nipping up the quill wouldn't have helped.
Mike Poole – thanks for your explanation of the relationship between helix rate and rake angle. Now you've pointed that out it seems obvious, but for me it was a wood and trees thing. Drills seemed so complicated I couldn't see it.
John Reese – thanks for the pointer to half round bits. I couldn't find them on MSC's UK site. Drill Service do them, but at £50 a pop that isn't going to happen. Are these like D-bits but with a 118 degree tip rather than a round end?
Since reading replies I have stoned the tips of the rubbishy Screwfix drills (which have been sitting on the shelf for 10+ years ) and now they go though brass like butter. No grab even on break though. Never throw anything away!
I'm OK with quick and dirty for now – it works!
Robin