Same here Vic though I don't tap mine in just make sure the taper and seating are clean.
The cause of the chuck spinning is the bit sticking in the blank so either drilling too fast and overheating, especially in acrylic or not clearing the chippings often enough. Some materials need more than others. Only ever happened to me once and that was my fault but fully under control and never dangerous IMO.
How can the chuck snap off the mt Alex unless the tailstock has been withdrawn too far, that puzzles me a bit as surely you would stop the lathe immediately it started spinning then twist it back out by hand. I'm confident enough to usually hold the chuck though not everyone would agree with that practice.
I have an excellent Meddings pillar drill but now always drill on the lathe as I find it easier, quicker and more accurate. I always drill right through the blank just making sure I clear fully before the last few mm and no blow out result. Recent purchase of pen jaws makes it even easier.
We're all different though and the best method is the one we're most comfortable with.
EDIT:
Just re read Alex and noted it was the bit not the mt that broke but my reasoning still stands. 
