feat(client): use selected Game Room brand icon - #9

Node.js 20 is deprecated. The following actions target Node.js 20 but are being forced to run on Node.js 24: actions/checkout@v4, actions/setup-node@v4. For more information see: https://github.blog/changelog/2025-09-19-deprecation-of-node-20-on-github-actions-runners/