Simplify README and move examples to documentation site

[maxisbey]

From the V2 checklist in #1253:

Making the README simple - all the examples should be in the documentation.
Simplify/remove/(move to docs) the current examples. Check how FastAPI does it - they have a docs_src where they put all their code examples, and they actually run tests on every snippet in tests.

The README currently contains extensive inline examples that make it long and hard to scan. The goal is to:

1. Slim down the README to a concise overview: what MCP is, installation, a single minimal example, and links to full docs.
2. Move detailed examples to the documentation site, where they can be organized by topic and tested.
3. Consider a docs_src/ pattern (like FastAPI) where code examples live as actual Python files that are included in docs via markdown and tested in CI.
4. Evaluate inline snippet approaches (like pydantic-ai) as an alternative.

PR #2008 (docs restructure) is a precursor to this work.

Ref: Kludex's V2 checklist in #1253