We do maintain a few dozen private repositories on GitHub and have discussed "GitHub Issues" with other paying customers as well.

Some teams do use GitHub issues as it's easy to connect to PRs, link to specific lines of a certain commit and generally relate to the context of ongoing development. This also reduces the management time from copying over tasks to another PM system and switching between views all the time, or lacking certain features that GitHub provides.

We have considered using GitHub issues actively for the development department and I can totally agree that more powerful GitHub issues with better categorization, custom fields, a bit more flexible views/reports and "Due Date" would help immensely for technical teams using private repos as well.