[quanticle]

    But would management approve of spending time on reporting and possibly
    fixing design issues? Well why not because if the knowledge of the issues
    exists only in the head of some developers, it is not really owned by the
    company. Whereas if it was reported in an issue-tracking system it would
    become official part of the IP owned by the company, increasing its value.

In practice, management will not endorse developers reporting and spending time working on design issues, unless that design issue can be directly identified as a blocker impeding the delivery of urgently needed functionality. Even then, half the time the developer will be told to note the issue and implement a short-term workaround.

    Using such a system would seem to make a lot of sense to me. Perhaps
    companies are already doing it?

Pretty much every company I've worked for has had a category in JIRA for design issues and technical debt. Tracking these things isn't the problem. The problem is getting buy-in to spend a portion of each sprint actually addressing these issues, rather than having them in some backlog pile that everyone pays lip service to, but no one takes action on actually reducing.