Understand what you're saying, though the point of a constant descent approach is to hit the MDA on profile (ie 2w/2r on PAPI), so if you can't get in off the CDA because there is cloud on profile at the MDA then you shouldn't be getting in off the dive and drive either if, as you say, you are not leaving the MDA below or above profile.
oh, and just in case...sometimes, in odd situations, a non precision apch may get you in while an ILS leaves you in the cloud for the same runway...
Not in our company it wouldn't, unless the ILS is not aligned with the runway. The final 400 feet we have to be stable including being on runway centerline. Cloud on the ILS at the DA would mean cloud on the final approach of the NPA as well.