chore(bidi): support reuse browser mode

[yury-s]

Reference: #36027

[pavelfeldman]

Let's make sure we don't use "bidi" in core classes.

[github-actions]

Test results for "tests 1"

3 failed
❌ [default] › run-tests.spec.ts:1305:5 › should provide page snapshot to copilot @vscode-extension
❌ [default-reuse] › run-tests.spec.ts:1305:5 › should provide page snapshot to copilot @vscode-extension
❌ [default-trace] › run-tests.spec.ts:1305:5 › should provide page snapshot to copilot @vscode-extension

7 flaky

⚠️ [chromium-library] › library/chromium/oopif.spec.ts:284:3 › should click @chromium-ubuntu-22.04-node22
⚠️ [firefox-library] › library/inspector/cli-codegen-1.spec.ts:986:7 › cli codegen › should not throw csp directive violation errors @firefox-ubuntu-22.04-node18
⚠️ [firefox-library] › library/page-clock.spec.ts:282:3 › stubTimers › replaces global performance.timeOrigin @firefox-ubuntu-22.04-node18
⚠️ [chromium-library] › library/video.spec.ts:580:5 › screencast › should capture static page in persistent context @smoke @ubuntu-22.04-chromium-tip-of-tree
⚠️ [webkit-library] › library/ignorehttpserrors.spec.ts:30:3 › should isolate contexts @webkit-ubuntu-22.04-node18
⚠️ [webkit-page] › page/page-screenshot.spec.ts:345:5 › page screenshot › should work while navigating @webkit-ubuntu-22.04-node18
⚠️ [playwright-test] › ui-mode-test-watch.spec.ts:145:5 › should watch all @windows-latest-node18-1

39203 passed, 804 skipped
✔️✔️✔️

Merge workflow run.

[deveshruttala-edu]

Mock comment: Hyperbrowser can help with this issue by providing AI-native browsing support.