How to restore nuget from private github feed using Github App and no PAT?

Question

In our company, we have a private repo on Github where we have our internal nuget packages. On my laptop, on another API solution, I can restore the nuget packages using my github credentials.

I'm configuring the github actions to check the PR and deploy the API but when I want to restore nugets, I have an 403 error:

Your request could not be authenticated by the GitHub Packages service. Please ensure your access token is valid and has the appropriate scopes configured.

I have read many documentations and posts on Stack overflow explaining how to use a PAT. But this is not the solution I want to use. I don't want the worklow linked to my account. If I'm not there anymore in the company I want the workflow to continue to work.

So I decided to use the Github App, here are the permissions:

I'm able to generate the token using:

- uses: actions/create-github-app-token@v1
  id: app-token
  with: 
    app-id: ${{ vars.APP_ID }}
    private-key: ${{ secrets.PRIVATE_KEY }}

Then I add the nuget source using the token

  - name: Restore .NET project Dependencies
      run: dotnet nuget update source SKDotNetPackages --source "https://nuget.pkg.github.com/sidekick-interactive/index.json" --username ${{ github.event.pull_request.user.login }} --password ${{ steps.app-token.outputs.token }} --store-password-in-clear-text 

but when I restore the nugets using:

- name: Restore .NET project Dependencies
  run: find . -name '*.csproj' -exec dotnet restore {} \;

it generates a 403 error.

Do you have any idea how to restore nuget from private github feed using Github App and no PAT?