Allow Model Picker Search Across All AI Providers

1) Global Search Across All Providers

Currently, the search function in the model picker popover only searches models from the selected AI provider. It would be much more convenient if it searched across all configured providers at once. This way, users wouldn’t need to first select a provider before searching. Instead, they could simply type the model name and instantly see matching results across all providers, making the selection process faster and more intuitive.

For example, I have multiple providers set up: OpenAI, Google AI, Anthropic, xAI, Perplexity, and OpenRouter. As it stands, I must first choose a provider and then search within it to find a specific model. A global search would allow users to simply type a model's name and see all matching results instantly, which would save time by removing extra clicks and streamlining the workflow.

2) Auto-Focus the Search Box

Bonus suggestion: It would also be great if the search box was automatically selected and focused as soon as the model picker is opened, so users can start typing immediately without an extra click on the search box.