While there may be an underlying consensus of mental instability, I strongly support not labeling it explicitly due to specific mental illness because while they may have DSM label "X", not all DSM label "x" are mass shooters, as you said here

"Politicians and journalists saying these things aren't due to mental illness are telling white lies in order to not promote stigma against the mentally ill, this I believe is the right thing to do however I believe it is completely incorrect"

Where I think you are wrong, and by wrong I mean a distinction should be made, is the argument of nature vs. nurture. I believe mass shooters are a combination of both, but the largest factor is nurture. This is because, in the example of psychopaths, imagine two identically people with stereotypical psychopathic tendencies. A well raised stable person may just have these be interpreted as personality quirks, while your isolated, misfit does not have the same opportunities to adapt and 'blend in' and may go on to be your next school shooter (or any type of shooter).

While this is a specific example, similar parallels can be made for many mental illnesses with changing parameters that allow for their integration with general population