api: Use Huma to generate swagger docs and parse JSON automatically (#262)

## Motivation and Context

Currently there's a lot of unsafe manual JSON unmarshalling/marshalling,
and we have to manually keep the openapi schema in sync.

This PR migrates the API to
[Huma](https://github.com/danielgtaylor/huma), which appears like a
popular, well-maintained way to build APIs in Golang. I'm not a Golang
expert, but seemed highly recommended by Reddit + Claude and has a
decent star count. It provides automatic openapi spec generation and
docs hosting, JSON parsing, as well as the ability to add auth
middlewares etc.

## How Has This Been Tested?

Running locally, and poking at the endpoints

## Breaking Changes

Docs are served at `/docs` rather than `/v0/swagger/index.html` now

## Types of changes
<!-- What types of changes does your code introduce? Put an `x` in all
the boxes that apply: -->
- [ ] Bug fix (non-breaking change which fixes an issue)
- [x] New feature (non-breaking change which adds functionality)
- [x] Breaking change (fix or feature that would cause existing
functionality to change)
- [x] Documentation update

## Checklist
<!-- Go over all the following points, and put an `x` in all the boxes
that apply. -->
- [x] I have read the [MCP
Documentation](https://modelcontextprotocol.io)
- [x] My code follows the repository's style guidelines
- [x] New and existing tests pass locally
- [x] I have added appropriate error handling
- [x] I have added or updated documentation as needed

## Additional context
<!-- Add any other context, implementation notes, or design decisions
-->

docs/faq.md

@@ -18,7 +18,7 @@ The MCP Registry is the official centralized metadata repository for publicly-ac
 There are four underlying concepts:
 - "MCP Server Registry API" (or "MCP Registry API"): The OpenAPI specification defined in [openapi.yaml](./server-registry-api/openapi.yaml). This is a reusable API specification that anyone building any sort of "MCP server registry" should consider adopting / aligning with.
 - "Official MCP Registry" (or "MCP Registry"): The application that lives at `https://registry.modelcontextprotocol.io`. This registry currently only catalogs MCP servers, but may be extended in the future to also catalog MCP client/host apps and frameworks.
-- "Official MCP Registry API": The REST API that lives at `https://registry.modelcontextprotocol.io/api`, with an OpenAPI specification defined at [swagger.yaml](../internal/docs/swagger.yaml)
+- "Official MCP Registry API": The REST API served at `https://registry.modelcontextprotocol.io`, which is a superset of the MCP Registry API. Its OpenAPI specification can be downloaded from [https://registry.modelcontextprotocol.io/openapi.yaml](https://registry.modelcontextprotocol.io/openapi.yaml)
 - "MCP server registry" (or "MCP registry"): A third party, likely commercial, implementation of the MCP Server Registry API or derivative specification.
 
 ### Is the MCP Registry a package registry?

docs/server-registry-api/README.md

@@ -10,4 +10,4 @@ The centralized, publicly available catalog ("Official MCP Registry") needs addi
 
 References:
 - [openapi.yaml](./openapi.yaml) - A reusable API specification (MCP Server Registry API) that anyone building any sort of "MCP server registry" should consider adopting / aligning with.
-- [swagger.yaml](../../internal/docs/swagger.yaml) - The specification backing the Official MCP Registry; a derivative of the MCP Server Registry API specification.
+- [https://registry.modelcontextprotocol.io/openapi.yaml](https://registry.modelcontextprotocol.io/openapi.yaml) - The specification backing the Official MCP Registry; a derivative of the MCP Server Registry API specification.