Why don't you think it should be part of product?
Love that your department is called "Quality", implies more metrics-less-feeling before reading the doc.
It's weird to me that the software engineers building the product are not viewed as "product folk".
The people designing the architecture for and building the actual product aren't managing the changes to it or concerned about it? The people who have to address any defects in it aren't concerned about their processes as they pertain to quality?
This sort of worldview seems anti-agile and anti-lean TBH.
If engineers aren't owning and focused on quality the business either doesn't really believe there are quality issues or there is a huge alignment issue.