Out of my experience with great QA in the past the two significant things they bring which poor QA does not are: 1) they create a detailed testing plan that allows them to 'checkbox' off things rather than simply throw time at randomly fucking around. 2) they have very rigorous reporting of issues including detailed repro steps, environment information, issue categorization, and (if the technology is provided) stack dumps.
Perhaps a company that provides this service specifically would work? Especially if the company had a technology aspect that allowed them to automate their own testing. Testing as a partnership service sounds like a business idea that makes great sense to me yet would be a difficult sell. It's rare to acknowledge that defects are guaranteed to exist, that a systematic plan to discover them can be created, that the earlier the discovery is found the lower the cost (exponentially) is to fix.
On the other hand those great QA of my past also had very strong domain knowledge of what they were testing. The audit step to create the plan would be important: the best QA really need to know the intended results for everything inside and out.
It all feels like a pipe dream to me, unfortunately. The norm seems to be "we don't want to invest in Quality Assurance, but we want Assured Quality, so just don't make mistakes!"