Doing every quality activity "after the fact" I agree is the issue. That's the root of the problem you're seeing, not that there was a separate quality team.
It’s not the “separate” part that I think is ridiculous. It’s the fact that the team is named “quality assurance.” It relies on a metaphor from
manufacturing that’s entirely inappropriate for software.
If you want to call it “Testing and Exploration” you’d get no argument from me. (Though I do think you’ll find that team is hard to staff.)