fix(webui): refresh skill suggestions when opening the picker

[Re-bin]

Summary

Fixes NAN-108.

Skills installed by the agent after the page loads can already be invoked by name, but the WebUI picker keeps its previous list. Users should not have to reload the page to discover a newly installed skill.

• Refresh the shared skill list whenever the $ picker opens, including when the cached list is empty.
• Keep filtering local while typing. Allow at most one request in flight, and coalesce opens during an older request into one trailing refresh so a pre-installation response cannot swallow the update.
• Preserve existing suggestions on transient failures and retain protection against stale responses overwriting newer settings updates.
• Cover post-load discovery, Tab completion, removals, trailing refreshes, listener cleanup, and retries.

No changes to skill installation, availability filtering, backend invocation, persisted data, or public APIs.

Verification

• Confirmed the initial discovery regression test fails before the fix and passes afterward.
• Review reproduced a second race: opening the picker during an older request dropped the refresh. The strengthened regression fails on the original PR and passes with the trailing-refresh fix.
• bun run test src/tests/skills-marketplace.test.tsx src/tests/thread-composer.test.tsx src/tests/app-layout.test.tsx — 161 passed on the updated branch.
• bun run lint and bun x tsc -p tsconfig.build.json — passed on the updated branch.
• Initial production build passed; current-head production build is also covered by the WebUI CI job.
• git diff --check — passed.