[IDEA] Use `<dialog>` and `.showModal()` for modals

[yaisog]

The <dialog> element was designed for this. It can be put anywhere in the DOM, being hidden, and gets lifted to the browsers top rendering layer with .showModal(). There is automatic background dimming, focus trapping, ESC key handling and more.
A drawback is that some browsers (Firefox, Safari) did not support this before 2022, so it doesn't match our baseline.

I made a demo PR #9587 using it, which also gets rid of the rigid header-body-footer modal structure in place of the template / cascade-based system (which isn't very polished, yet) for testing and feedback.