chore: align mcp-extension build and deps with the monorepo

[dgozman]

No description provided.

[github-actions]

Test results for "tests 1"

4 flaky

⚠️ [chromium-library] › library/chromium/oopif.spec.ts:284 › should click `@chromium-ubuntu-22.04-node22`
⚠️ [firefox-library] › library/inspector/cli-codegen-1.spec.ts:1079 › cli codegen › should not throw csp directive violation errors `@firefox-ubuntu-22.04-node18`
⚠️ [firefox-page] › page/page-event-request.spec.ts:182 › should return response body when Cross-Origin-Opener-Policy is set `@firefox-ubuntu-22.04-node18`
⚠️ [firefox-page] › page/page-wait-for-function.spec.ts:104 › should work with strict CSP policy `@firefox-ubuntu-22.04-node18`

46737 passed, 819 skipped

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Merge workflow run.

[yury-s]

See #37313