integration tests: Expose and use CodeMirror API to read editor value

Created by: fkling

Reading the value directly from the DOM nodes is unreliable since we don't necessarily know which decorations are displayed which could impact the return value.

Using the work from #38805 this exposes CodeMirror's own API to get an editor instance from the DOM and read the real value from that API.

Test plan

INTEGRATION_TESTS=1 yarn watch-web
yarn _test-integration client/web/src/integration/search.test.ts

App preview:

• Web
• Storybook

Check out the client app preview documentation to learn more.