I think pragmatism is a fine approach here, but could you clarify for me what your think the answer to my question is exactly? If it's not meaningful to ask whether or not there are indivisible and discontinuous units, then is the answer to my question "Does QM's claims about Planck time imply that time is discontinuous?" simply "No" because QM says nothing meaningful about the question one way or the other?
In ‘pure’ QM (without gravity), the Planck length has no special significance, and spacetime is assumed to be continuous. But we know that QM as we know it must be an approximation because it disagrees with GR (and/or vice versa), and the ‘correct’ theory of quantum gravity might predict weird things at the Planck scale. So far, most proposed theories of quantum gravity have little more predictive power than “The woman down the street is a witch; she did it”, though some do predict stuff such as the dispersion of gamma rays I've mentioned elsewhere.