When running the Designer as root on Debian, the Perspective node is missing from the Project Browser, as shown in the screenshot from my original post.
But when running the Designer as a normal user on Debian, the Perspective node appears normally and works as expected.
At this point I'm not sure whether the JxBrowser error is directly related, since the same error occurs during Vision startup and Vision still loads correctly.
The only reproducible difference I found is that the issue occurs when running the Designer as root, while running it as a normal user works correctly.
Since running the Designer as a normal user resolves the issue for me, I will continue using it that way for now.
We use an "embedded" web browser, called JxBrowser, which is ultimately Chromium. Chromium doesn't want to be run as root, for the sake of security, and actively prevents you from launching as root unless you jump through some hoops. Running the designer process as not-root is definitely recommended practice, for this and other reasons.