Geoff – the few times I have had to use stud extractors they have done the business for me, except I have now 'lost' them, or the 'Little People' have borrowed them, same difference really but if the latter I may get them back later!
From what you say it sounds like you did indeed drill the pilot hole too large so the broken steel stud expanded and seized tighter in the aluminium head. Had it been a stainless stud it would have seized in the aluminium anyway – on boats we had to use some special gunge between stainless fittings and the aluminium mast otherwise they were there for ever! Even mild steel, never mind stainless, into aluminium is fraught with the chance of seizure, hence your sheared stud. So, even if the stud extractor had not broke you might still have had a "no go" on your hands.
Best way to get a broken stud/bolt out if no easy-outs available to to drill a small pilot drill hole as accurately straight as possible then successively larger holes until the drill cuts the edge of the threads then, if you are really lucky, you can "persuade" the broken bit to collapse inwards and be pulled out as you have found out for yourself – sorry, granny and egg sucking comes to mind! Like Jason says, helicolis are an answer and the most reliable way of threading in aluminium if high loads and/or need to 'frequently' disassemble items are a requirement.
Chris