There is no correct response; in my experience, the QA team wrote all regression and smoke suites. You can't separate manual and automated testing and rely on just one. The BDD approach is appropriate for businesses where managers will read your Cucumber tests to clarify what is being tested and what should be tested; otherwise, the time will be wasted. I recommend that the QA TL meet with the PMs to discuss those points; following that, you may choose the best method for your team.
BDD is more than just a quality assurance tool. It is a procedure that the entire team follows.