Bot users don't have access to internal projects

Summary

A pipeline created with an project access token fails when including a .gitlab-ci.yml template from an internal project.

Steps to reproduce

1. Create a project with 'internal' visibility and create a file .gitlab-ci.template.yml to reference as a template.
2. Create another project. In this second project:

   1. Create a .gitlab-ci.yml which includes the template from the first project similar to this:

      include:
      - project: 'path/to/the/internal/project'
        file: '/.gitlab-ci.template.yml'

   2. Create a project access token with api and write_repository permissions.

3. Trigger a pipeline by pushing a commit using the access token.
4. Also trigger a pipeline as a "regular" user to see it not fail.

What is the current bug behavior?

The pipeline triggered by the bot user fails with

Found errors in your .gitlab-ci.yml:

• Project path/to/the/internal/project not found or access denied! Make sure any includes in the pipeline configuration are correctly defined.

What is the expected correct behavior?

The pipeline is started just like as it would be by a logged in user. Note many reports were not reproducible due to incorrect identification of private vs internal projects however #361993 (comment 1674185875) may be helpful in reproduction.