fix(html): remove filter clear button

[Skn0tt]

Closes https://github.com/microsoft/playwright/issues/33681. We tried fixing the nav behaviour in https://github.com/microsoft/playwright/pull/33689, but couldn't come up with behaviour that works well with the browser history. See https://stackoverflow.com/questions/2977023/how-do-you-detect-the-clearing-of-a-search-html5-input for why we cannot detect if the x was clicked or the text was deleted.

To remove confusion, this PR removes the clear button. Users can still clear with ⌘+a 🔙, and it'll do the right thing.

[github-actions[bot]]

Test results for "tests 1"

1 flaky

:warning: [playwright-test] › ui-mode-test-watch.spec.ts:145:5 › should watch all @windows-latest-node18-1

37173 passed, 650 skipped :heavy_check_mark::heavy_check_mark::heavy_check_mark:

Merge workflow run.

[github-actions[bot]]

Test results for "tests 1"

1 failed :x: [playwright-test] › runner.spec.ts:118:5 › should ignore subprocess creation error because of SIGINT @macos-latest-node18-1

4 flaky

:warning: [installation tests] › playwright-electron-should-work.spec.ts:31:5 › electron should work with special characters in path @package-installations-macos-latest
:warning: [installation tests] › playwright-test-package-managers.spec.ts:54:5 › npm: uninstalling ct removes playwright bin @package-installations-macos-latest
:warning: [webkit-library] › library/screenshot.spec.ts:219:14 › element screenshot › element screenshot should work with device scale factor @webkit-ubuntu-22.04-node18
:warning: [playwright-test] › ui-mode-test-watch.spec.ts:145:5 › should watch all @windows-latest-node18-1

37245 passed, 650 skipped :heavy_check_mark::heavy_check_mark::heavy_check_mark:

Merge workflow run.