fix(ci): restore source HTML link checks

[drewstone]

Summary of changes

• Restore link checks for the three source index.html files omitted by merged chore: bump lycheeverse/lychee-action from 2.5.0 to 2.9.0 #3319.
• Mirror each app's Vite served root in runner scratch space: public/ files at / and src/ at /src.
• Run Lychee in offline mode against each HTML entrypoint after the existing Markdown and reStructuredText check. The existing action still checks network links in those document formats.

Proposed area of change

CI workflow only. No application source or published artifact changes.

Associated issue(s)

Follow-up to merged #3319.

Verification

• Extracted and ran the checked-in workflow script with Lychee 0.24.2, the version installed by lychee-action@v2.9.0: 3 local links passed in each of the three source HTML files.
• Injected a missing /favicon.ico target into a copy of apps/tangle-cloud/index.html. The checked-in script failed with exit 2 and identified the missing path and source line.
• Workflow YAML parsed, Prettier and git diff --check passed.
• The repository pre-push hook passed lint, format, tests, and build.

Screen Recording

No visible product change.

Harness Validation (Required for Launch-Flow Impact)

No launch flow changes; wallet harness is not applicable.

[tangletools]

✅ Auto-approved PR — bb480dae

Blanket team auto-approval is intentional. The merge gates are CI and codex-p1.
No automated review runs on this PR. This approval rests on the rule above alone.

_{tangletools · auto-approval · reason: blanket_auto_approve · 2026-09-30T00:09:01Z}