If correct, then the structure of that symbiotic union is part of what is passed on to the next generation, and the way that structure is passed on is not as part of the DNA specification.
I'm not sure quite how you work this out. All the information required to produce the various components for any structure you care to mention, mitochondria, chloroplasts, nuclear membranes or cilia are all encoded in the DNA either in the nucleus or between the nucleus and the discrete genetic complements of the organelles in the case of mitochondria and chloroplasts. Apart from the initial cytoplasmically or membrane inherited organelles any organelle is going to be a product of the DNA specification shared between the nuclear and organellar DNA.
Are you saying that any protein inherited cytoplasmically during mitosis is not 'part of the DNA specification'?
But the persistence of the mitochondria in the cell is a result of the genetic complement involved in the synthesis of new mitochondria. A mitochondrion is not an autonomous genetic unit. While mitochondrial division resembles bacterial division it requires the input of a number of exogenous compounds from the host cell including proteins synthesised from nuclear genes of the host.
In what way is the persistence of the mitochodria in the cytoplasm not a result of the shared genetics of that organelle and the resulting structure and molecular character of the mitochondria a product of those genes.
The initial engulfment of the proto-mitochondrial bacteria and subsequent generations of solely cytoplasmic inheritance might fit your example but the current situation does not as far as I can see.