chore(deps): upgrade dependencies (DOCSDEV-69) (#669)

- astro 6.1.8 → 6.3.8 (pinned ~6.3.8; astro 6.4 incompatible with
  Starlight 0.39.2's pinned @astrojs/mdx ^5.x)
- @astrojs/starlight 0.38.3 → 0.39.2
- @astrojs/check 0.9.8 → 0.9.9
- @interledger/docs-design-system 0.12.0 → 0.13.0
- starlight-links-validator 0.23 → 0.24
- typescript 5.9.3 → 6.0.3 (MAJOR; lint + build pass; 4 pre-existing
  astro check errors are unchanged from main, not introduced by TS 6)
- eslint 9.39.4 → 10.4.0 (MAJOR), @eslint/js 9 → 10
- globals 17.5 → 17.6
- @typescript-eslint/parser, typescript-eslint 8.59 → 8.60

Linux-only optional deps (@pagefind/linux-x64, @rollup/rollup-linux-x64-gnu)
left untouched per existing CI pinning.

Author: Infi-Knight