refactor(stage-ui): use public package exports - #2373

Node.js 20 is deprecated. The following actions target Node.js 20 but are being forced to run on Node.js 24: autofix-ci/action@7a166d7532b277f34e16238930461bf77f9d7ed8. For more information see: https://github.blog/changelog/2025-09-19-deprecation-of-node-20-on-github-actions-runners/

packages/stage-ui/src/tools/web-search.ts:84:24: Unexpected control characters

packages/pipelines-audio/src/llm-streaming-control/controller.ts:191:30: Using a spread operator here creates a new array unnecessarily.

packages/pipelines-audio/src/managers/playback-manager.test.ts:146:9: Expected expression to be used

packages/pipelines-audio/src/managers/playback-manager.ts:398:24: Using a spread operator here creates a new array unnecessarily.

packages/pipelines-audio/src/managers/playback-manager.ts:385:26: Using a spread operator here creates a new array unnecessarily.

packages/pipelines-audio/src/managers/playback-manager.ts:371:26: Using a spread operator here creates a new array unnecessarily.

packages/pipelines-audio/src/managers/playback-manager.ts:67:29: Using a spread operator here creates a new array unnecessarily.

packages/stage-ui/src/libs/providers/validators/run.ts:96:106: Parameter '_' is declared but never used.

packages/stage-ui/src/libs/zod/index.ts:16:17: Parameter '_' is declared but never used.