Add default PostHog SDK client facade

[marandaneto]

💡 Motivation and Context

Adds an optional process-wide default PostHog client facade for console apps, scripts, and other places where passing an IPostHogClient instance around is inconvenient.

💚 How did you test it?

• dotnet build src/PostHog/PostHog.csproj --no-restore -v:minimal
• dotnet test tests/UnitTests/UnitTests.csproj --framework net8.0 --no-restore --filter PostHogSdkTests -v:minimal

📝 Checklist

• I reviewed the submitted code.
• I added tests to verify the changes.
• I updated the docs if needed.
• No breaking change or entry added to the changelog.

If releasing new changes

• Added the release label to the PR
• Added exactly one version bump label: bump-patch, bump-minor, or bump-major

[github-actions]

posthog-dotnet Compliance Report

Date: 2026-04-30 12:44:29 UTC
Duration: 221ms

⚠️ Some Tests Failed

0/1 tests passed, 1 failed

---

Feature_Flags Tests

⚠️ 0/1 tests passed, 1 failed

View Details

Test	Status	Duration
Request Payload.Request With Person Properties Device Id	❌	42ms

Failures

request_payload.request_with_person_properties_device_id

404, message='Not Found', url='http://sdk-adapter:8080/get_feature_flag'

[marandaneto]

@haacked wdyt? i noticed some SDKs don't have a global facade, and you have to pass it around, and sometimes passing it around is tricky, you have to change a lot of method and ctor signatures, and make your code even more coupled to PostHog