fix(example): stabilize Expo navigation and startup

[dennytosp]

Summary

• route the existing React Navigation app through Expo Router SDK 56+ compatibility exports
• fix TypeScript errors across the example source and add a repeatable typecheck command
• drive the Profile scroll-to-top arrow directly from the native scroll offset
• upgrade Bootsplash to v7, align native/React assets, and keep an opaque overlay until navigation and initial task loading settle

Validation

• bun run typecheck
• bun run install-check
• targeted ESLint on navigation, splash, task state, and Profile files
• successful iOS native build on iPhone 16 simulator
• cold-start QA with normal loading and a temporary 10-second task delay; splash stayed opaque until fulfilled/rejected readiness and then transitioned to Home

Notes

• react-native-bootsplash is intentionally excluded from Expo dependency validation because SDK 57 metadata still expects 6.x, while upstream v7 supports Expo SDK 54+ / React Native 0.80+
• unrelated local config and dependency bumps were left out of the commits