feat(infra): integrate PostHog for feature flags and analytics

Summary

Add PostHog as the feature flag and product analytics platform. Enables gradual rollout of the membership system, instant kill switches for auth, user-targeted beta access, and usage analytics — all in one open-source tool.

Requirements

Integration

• Install posthog-js and posthog-js/react
• Initialize PostHog provider in App.tsx (wrap router)
• Configure respect_dnt: true for Do Not Track compliance
• Environment variable for PostHog project key (VITE_POSTHOG_KEY)
• Disable in development by default

Feature Flags

• membership_enabled — master switch for the entire auth/profile system
• registration_open — controls whether new users can sign up
• admin_dashboard — gates access to the admin panel
• profile_public — controls whether member profiles are publicly visible
• Create a useFlag(name) hook wrapping PostHog's useFeatureFlagEnabled

Analytics Events (initial set)

• Page views (automatic via PostHog)
• user_signed_up — new member registration
• user_signed_in — sign-in event
• profile_updated — member edited their profile
• job_viewed — clicked a job listing
• group_joined — joined a working/affinity group

Privacy

• Add PostHog to the site's privacy/cookie notice
• Respect Do Not Track browser setting
• No PII in event properties (use anonymous IDs until user opts in)
• Document self-hosting option for future institutional compliance needs

Context

Parent issue: #1916
Should be integrated alongside issue #1918 (auth) since feature flags are most valuable during the auth rollout. PostHog's free tier (1M events/month) is sufficient for a 4,000-member community.

Implementation Notes

• Use PostHog Cloud initially; document self-hosting path for future if needed
• PostHog React SDK provides PostHogProvider, useFeatureFlagEnabled, and usePostHog hooks
• Feature flags evaluate on the client — wrap protected routes with flag checks
• For SSR/prerender compatibility, flags should have sensible defaults (off) while loading
• Consider PostHog session replay for debugging auth flows during beta

Start with App.tsx and the existing router, auth flows, and profile or membership entry points; then review the PostHog React SDK integration requirements in the issue. Done means the listed flags and events work, development is disabled by default, Do Not Track and no-PII requirements are respected, and the privacy notice documents the integration and self-hosting path.