fix: guard startup doSearch initialization

[Dodothereal]

Summary

• replace the unconditional doSearch() call in index.html with a bounded readiness check
• initialize search only after window.activity.doSearch and searchWidget are available
• avoid startup race conditions from script load order

Testing

• npm run lint
• loaded http://127.0.0.1:3000 with Playwright CLI and confirmed no doSearch is not defined startup error

Fixes #5313

[github-actions]

✅ All Jest tests passed! This PR is ready to merge.

[vanshika2720]

@Dodothereal A PR(#5314 ) already exits on this issue .

[Dodothereal]

Closing to avoid duplicate/overlapping work. I will only submit PRs for clearly unclaimed issues (no active PR and no contributor already working on it).