The tang is there for ejection not drive.
True. Once the tool is in and has had some pressure applied via the feed then it sticks and doesn't spin. (except for the sharp ones that grab and pull out).
But I did have problems starting a cut say, with a 15mm drill opening up a 12mm hole, where the tool would spin fairly reliably at the start. Drove me nuts.
The bed on my lathe is quite short, so 'slide the tool in with a bit of feeling' is trickier than it sounds..
My old drill press has a slot that the tang goes into, at least one of the lathes I've owned had a similar arrangement..
I suspect you are right though, the problem is an error in the taper.. I don't own an MT3 reamer. but I do own some grub screws.... Works for me..
PK