chore(auto-carousel): remove auto-carousel API route, component, and all references

[natashaannn]

User story

As a developer, I want the auto-carousel feature completely removed from the codebase, so that there are no orphaned API routes or dead components after the page is deleted in Issue #76.

Background

Issue #76 deletes app/auto-carousel/page.tsx (the route) and app/login/page.tsx as part of the route reorganisation. However, it explicitly excludes changes to app/api/ and app/components/. This issue completes the auto-carousel removal by deleting the backend API route and the form component, and cleaning up all references.

Auto-carousel functionality: The feature called the Claude API to analyze a YouTube video transcript (fetched via CaptionExtractor) and automatically suggest carousel content. This functionality is being removed — it is not being moved to another route or preserved elsewhere.

Acceptance criteria

Happy path

Given this issue is merged
When a developer runs grep -r "auto-carousel\|AutoCarousel" app/ --include="*.ts" --include="*.tsx"
Then zero results are returned

Given the app is running after this issue is merged
When a request is made to POST /api/auto-carousel
Then Next.js returns 404 (no handler exists)

Given the landing page (app/page.tsx) is viewed
When this issue is merged
Then no tool card or link to /auto-carousel is present

Edge case

Given Header.tsx contains a nav link to /auto-carousel
When this issue is merged
Then that nav link is removed and no broken link remains

Out of scope

• Deleting app/auto-carousel/page.tsx itself (handled in Issue refactor(app): reorganise routes into (public) and (internal) Next.js route groups #76)
• Changes to the /carousel or /get-youtube-captions routes
• Removing the ANTHROPIC_API_KEY env var (it may still be used by other future features)

Technical context

Files to DELETE:

• app/api/auto-carousel/route.ts
• app/components/AutoCarouselForm.tsx

Files to EDIT (remove auto-carousel references):

• app/page.tsx — remove the Auto Bulk Carousel tool card and any link to /auto-carousel
• app/components/Header.tsx — remove the Auto Carousel nav link if present
• app/login/page.tsx — this file is also deleted in Issue refactor(app): reorganise routes into (public) and (internal) Next.js route groups #76; if that issue has already landed, no action needed here

Verify no other files import AutoCarouselForm or reference /auto-carousel:

grep -r "AutoCarouselForm\|auto-carousel" app/ --include="*.ts" --include="*.tsx" -l

Implementation details

1. Delete app/api/auto-carousel/route.ts
2. Delete app/components/AutoCarouselForm.tsx
3. In app/page.tsx: remove the Auto Bulk Carousel / Pro feature tool card and its import
4. In app/components/Header.tsx: remove the Auto Carousel nav entry if present
5. Run npm run build to confirm no import errors remain

Additional test scenarios

• npm run build completes with zero errors after deletions
• tsc --noEmit passes — no dangling imports referencing the deleted component or route types

Hard constraints

• Do not replace the auto-carousel feature with a stub or redirect — delete it entirely
• Do not touch scripts/carousel/CaptionExtractor.js or CarouselGenerator.js (used by /carousel)
• Do not change the ANTHROPIC_API_KEY env var handling in other parts of the app

Dependency issues

• Should be merged after Issue refactor(app): reorganise routes into (public) and (internal) Next.js route groups #76 so the page directory deletion is already done and file paths are consistent
• Independently mergeable from Issues feat(auth): add modular Resend subscriber email gate for carousel and get-youtube-captions #77, feat(api): add IP rate limiting to public carousel and youtube-captions API routes #78, feat(api): add secret token guard to internal filesystem-writing API routes #79