This is starting to reach to the edges of what I am sure about, but I will answer the question as best I can, and hope I don't mislead you!
The stretch correction process certainly indicates that there is some physical mechanism in the Supernovae explosion that leads to this correlation. I believe there is some relationship between the stretch correction and properties of the SN spectra, in particular the strength of some metal lines, particularly Nickel. The strength of the line tells us how much of that metal is present in the SN progenitor (the thing that the SN is before it goes boom) and I think it is the different metal abundance that regulates the peak brightness and width. This is another reason to doubt that high redshift SN1A are different from low redshift ones, since we can observe these abundances in both cases.