There is a missing step in your main.yml that's why your workflow can't find and access the files. To fix it, add checkout@v2 action in your steps:

name: Deployment

on:
  push:
    branches:
    - master
  
jobs:
  deploy:
    name: Deploy to App Engine
    runs-on: ubuntu-latest
    steps:
      - name: Checkout
        uses: actions/checkout@v2
      - name: Google App Engine
        uses: google-github-actions/[email protected]
        with:
          project_id: my-gae-project-id
          version: master
          credentials: ${{secrets.GOOGLE_APP_ENGINE_KEY}}
          flags: --no-cache
          deliverables: app.yaml

You can also visit this official github example for your reference.

Looks like the original push-to-deploy feature is now deprecated, but you can use Google Cloud Platform's Build Trigger to do this:

Navigate to Google Cloud Platform > Container Registry > Build Triggers and set up the branch(es) you want to auto build from your connected github repository.

Make sure you've added a build definition to your repository. Here you can find the full specification, but here's an example of the bare minimum to do a gcloud deploy via cloudbuild.yaml:

steps:
- name: 'gcr.io/cloud-builders/gcloud'
  args: ['app', 'deploy']