migrate dart_mcp to package:checks

[jakemac53]

Migrates dart_mcp to package:checks as a test case.

This is a pure AI migration in the initial commit.

Notable changes

• Exposed a new api.dart public library which contains no server or client specific APIs.
• Made all extension types implement Object, this helps with writing extensions on these extension types, otherwise they all appear nullable and checks extensions don't really work.

[github-actions]

PR Health

Changelog Entry ❗

Package	Changed Files
package:dart_mcp	pkgs/dart_mcp/lib/api.dart pkgs/dart_mcp/lib/client.dart pkgs/dart_mcp/lib/server.dart pkgs/dart_mcp/lib/src/api/api.dart pkgs/dart_mcp/lib/src/api/sampling.dart pkgs/dart_mcp/pubspec.yaml

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