The classic CLI run() loop calls prewarm_picker_cache_async() during the
idle window after the banner is shown, so the first /model open hits a warm
provider-models disk cache and renders in ~100ms. The stdio TUI entry point
never did this, so the first /model open in a TUI session blocked on serial
/v1/models fetches for every authenticated provider.
Mirror the CLI behaviour: kick off the same off-thread prewarm right after
gateway.ready is emitted (banner shown, user about to type). Fire-and-forget,
guarded once-per-process, fully exception-isolated so a slow or offline
provider can never affect TUI startup.
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