test: add example app verification

[joker23]

Note

Medium Risk
Moderate risk due to CI workflow changes that assume AWS roles and fetch secrets, which can affect pipeline security and stability. App/runtime changes are minimal and limited to reading an env-provided flag key.

Overview
Adds a new run-example GitHub Actions job that builds the React client-only example, pulls required SDK/flag keys from AWS SSM via OIDC, and runs Playwright e2e tests; the workflow also gains a nightly schedule trigger and skips the existing matrix build on scheduled runs.

Introduces Playwright test infrastructure for the example app (config + verify.spec.ts), wires a test script and @playwright/test dependency, ignores test-results/, and updates the app to read the feature flag key from import.meta.env.LAUNCHDARKLY_FLAG_KEY (with a fallback).

^{Written by Cursor Bugbot for commit 8c5c607. This will update automatically on new commits. Configure here.}

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[github-actions]

@launchdarkly/js-sdk-common size report
This is the brotli compressed size of the ESM build.
Compressed size: 25566 bytes
Compressed size limit: 26000
Uncompressed size: 125383 bytes

[github-actions]

@launchdarkly/browser size report
This is the brotli compressed size of the ESM build.
Compressed size: 172130 bytes
Compressed size limit: 200000
Uncompressed size: 800872 bytes

[github-actions]

@launchdarkly/js-client-sdk size report
This is the brotli compressed size of the ESM build.
Compressed size: 24212 bytes
Compressed size limit: 25000
Uncompressed size: 83755 bytes

[github-actions]

@launchdarkly/js-client-sdk-common size report
This is the brotli compressed size of the ESM build.
Compressed size: 21281 bytes
Compressed size limit: 24000
Uncompressed size: 110213 bytes

[joker23]

@cursor review

[devin-ai-integration]

✅ Devin Review: No Issues Found

Devin Review analyzed this PR and found no potential bugs to report.

View in Devin Review to see 3 additional findings.

[cursor]

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