chore(ui): Dialog UI for upcoming settings menu

[agg23]

The dialog/new UI controls from #34010, separated into its own PR.

This change only touches the settings view in UI Mode, which has been verified to continue functioning as normal.

[github-actions]

Test results for "tests 1"

8 flaky

⚠️ [firefox-page] › page/page-evaluate.spec.ts:403:3 › should throw for too deep reference chain @firefox-ubuntu-22.04-node18
⚠️ [installation tests] › playwright-test-package-managers.spec.ts:54:5 › npm: uninstalling ct removes playwright bin @package-installations-macos-latest
⚠️ [webkit-library] › library/page-event-crash.spec.ts:42:1 › should throw on any action after page crashes @webkit-ubuntu-22.04-node18
⚠️ [webkit-library] › library/page-event-crash.spec.ts:55:1 › should cancel waitForEvent when page crashes @webkit-ubuntu-22.04-node18
⚠️ [webkit-library] › library/selector-generator.spec.ts:197:5 › selector generator › should use internal:has-text @webkit-ubuntu-22.04-node18
⚠️ [webkit-library] › library/trace-viewer.spec.ts:109:1 › should show tracing.group in the action list with location @webkit-ubuntu-22.04-node18
⚠️ [webkit-page] › page/page-set-input-files.spec.ts:147:3 › should upload large file @webkit-ubuntu-22.04-node18
⚠️ [playwright-test] › ui-mode-test-watch.spec.ts:145:5 › should watch all @windows-latest-node18-1

37389 passed, 650 skipped
✔️✔️✔️

Merge workflow run.