fix: various WebSocketRoute issues

[dgozman]

• Do not allow two clients to route simultaneously.
• Cleanup after dispatcher disconnects.
• Ensure that routing works in context reuse scenarios.
• Test context reuse both in the launched browser with a single connection, and in a connected browser with different connections.

Fixes #36480. References #35987.

[github-actions]

Test results for "tests 1"

8 flaky

⚠️ [chromium-library] › library/chromium/tracing.spec.ts:49:3 › should run with custom categories if provided @chromium-ubuntu-22.04-node20
⚠️ [firefox-library] › library/inspector/cli-codegen-1.spec.ts:1087:7 › cli codegen record › should not throw csp directive violation errors @firefox-ubuntu-22.04-node18
⚠️ [firefox-page] › page/page-event-request.spec.ts:182:3 › should return response body when Cross-Origin-Opener-Policy is set @firefox-ubuntu-22.04-node18
⚠️ [playwright-test] › ui-mode-test-watch.spec.ts:145:5 › should watch all @ubuntu-latest-node18-1
⚠️ [playwright-test] › ui-mode-test-watch.spec.ts:145:5 › should watch all @ubuntu-latest-node22-1
⚠️ [webkit-library] › library/inspector/pause.spec.ts:74:5 › pause › should pause and resume the script with keyboard shortcut @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

47188 passed, 980 skipped
✔️✔️✔️

Merge workflow run.