Fix staging workflow

[hsubox76]

• Forgot to remove push trigger
• Update a couple of things so that the github-script merge action will work (see https://github.com/firebase/firebase-js-sdk/blob/master/.github/workflows/merge-release-branch.yml for a working merge workflow)

[changeset-bot]

⚠️ No Changeset found

Latest commit: 7df712c

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[google-oss-bot]

Size Report ¹

Affected Products

No changes between base commit (af7a182) and merge commit (a05887c).

Test Logs

• Base (af7a182): https://github.com/firebase/firebase-js-sdk/actions/runs/2374205238
• Merge (a05887c): https://github.com/firebase/firebase-js-sdk/actions/runs/2379311613

1. https://storage.googleapis.com/firebase-sdk-metric-reports/wSzt6fdMVv.html

[dwyfrequency]

LGTM

[google-oss-bot]

Size Analysis Report ¹

Affected Products

No changes between base commit (af7a182) and merge commit (a05887c).

Test Logs

• Base (af7a182): https://github.com/firebase/firebase-js-sdk/actions/runs/2374205238
• Merge (a05887c): https://github.com/firebase/firebase-js-sdk/actions/runs/2379311613

1. https://storage.googleapis.com/firebase-sdk-metric-reports/7w8L4PMEom.html