My most successful client (a company I have ownership in, now) has another slot besides marketing, design and code. That is our point person for filtering, testing and verifying user experience issues. Putting a single point person in charge of that onslaught of emails, who fully understands the software, and having them run a full time bug reporting/feature request channel, I think, is indispensible. They advocate on behalf of users but also know when the requests are silly or something is user error. They know whether an issue is mainly design or mainly code. Having them in place and engaging daily with users means we can filter out 90% of the noise in the signal. But it needs to be coupled with open-mindedness to customer feedback from the earliest iterations. Whatever requests or issues come through that person must be addressed. That person should understand what they can and cannot promise customers, what is crucial vs what is fluff, and how to prioritize those requests.
Her official role is "General Manager" but in fact she was promoted from a customer service role and the position was created for her because she was so good at spending extra time off-hours writing detailed, reproducible bug reports on behalf of customers who had experienced some issue. Reproducing and screenshotting the flow and the issues herself.
This person is a 10x force multiplier by virtue of being a power-user of the software who also interacts with customers and management daily, although she has no code or design experience.