So I hacked on https://inputbuffer.io and just opened it to a wider audience.
You hook up your user feedback source (via widget or API) and it will organize everything by content category (e.g. billing) or target (e.g. a specific page, API endpoint, CLI command etc).
Categorization isn't rigid, InputBuffer does its best to put feedback where it belongs and gives you a clear triage flow if you want the added control.
Once organized you can learn more via a quick analytics dashboard or by interrogating the data directly, chatting with InputBuffer to gain a stronger understanding of your product, with clear citations to all feedback.
I have had success on both small and large amounts of input, on traditional SaaS platforms, developer tools, open source projects and more.
Next up: automatically gathering user input from other platforms (like GitHub issues), and more research tools.