Refactor custom header after Starlight minor version bump

[huijing]

The latest version of Starlight (https://github.com/withastro/starlight/releases/tag/%40astrojs%2Fstarlight%400.11.0) released its long-awaited feature for customising Starlight components. Thus far, we have been using a community-sourced "hack" method of doing this (see https://gist.github.com/giuseppelt/7f918a3ac02a011d76811ae472f8bf09), with the intention of migrating to the "official" method of component overriding the moment it was baked. The time has come for this to happen.
We will also need to make some styling tweaks and checks because the API of how the Starlight internals (i.e. CSS classes, HTML structure) would have changed a bit due to the additional feature.