docs: expand AI Providers index to include full provider directory

[Olusammytee]

Summary

• reorganize the AI Providers landing page into clearer provider categories
• add first-party, foundation model, gateway, cloud platform, and local/runtime sections
• include a complete A-Z provider directory for quick navigation

Why

This makes provider discovery faster and gives users a single index to navigate the full provider set.

Migrated from Kilo-Org/kilocode#5960 due docs repo migration.

[Copilot]

Pull request overview

This PR reorganizes the AI Providers landing page to improve provider discovery and navigation. It expands from a simple categorized list with descriptions to a comprehensive directory with clearer organization and an alphabetical index.

Changes:

• Reorganized providers into 5 focused categories: First-Party Access Options, Foundation Model Providers, Gateways and Compatible APIs, Cloud Platform Integrations, and Local and Editor-Hosted Models
• Expanded coverage from ~12 to 34 providers with consistent linking structure
• Added a complete A-Z provider directory for quick reference navigation

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[Copilot]

The provider name is inconsistent with its documentation page. The provider page (gemini.md) uses "Google Gemini" as the sidebar_label and throughout the documentation, but here it's listed as just "Gemini". For consistency and clarity, this should be "Google Gemini" to match the other references, or alternatively, all references including line 29 should use "Gemini" without the "Google" prefix.

Suggested change

	- [Gemini](/docs/ai-providers/gemini)
	- [Google Gemini](/docs/ai-providers/gemini)