RE: your update - I have several brands of dies here - so far as I know, only LEE uses the collet to hold the expander shaft - the others here - RCBS, Lyman, Bonanza, etc, all are threaded - so more "solid", but also, less "forgiving" of user error - set up has to be correct - follow the instructions. Each seems to have a different shape to the actual expander ball - depending what you are wanting to do, can make a difference - I like the Lee's long taper, or the Hornady elliptical shape, for forming one brass size into a different brass size. However, I have loaded literally thousands of rounds with the RCBS dies - almost no issues at all, unless operator error in setting them up - usually get something bent or broke - then look at instructions and pretty apparent to me what I did wrong.
I can't help any more about the slipping that you are getting with the Lee collet system - it has its advantages, but maybe you have found its weaknesses??