chore: push browser to the client when launching persistent

[Dmitry Gozman (dgozman)]

This makes context.browser() to not be null.

References #35987.

[github-actions]

Test results for "tests 1"

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

6 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-page] › page/page-wait-for-function.spec.ts:104:3 › should work with strict CSP policy @firefox-ubuntu-22.04-node18
⚠️ [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

39210 passed, 804 skipped
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Merge workflow run.