@MYROLD Mathias Thanks for reaching out to Microsoft Q&A, apologize for any inconvenience caused on this.
I have followed the same documentation steps & able to create and request the static web apps successfully.
You can monitor whether the deployment has been triggered to static webapp or not using the Jobs --> under Build --> under your GitLab Repo as shown below.
Also, modify the property "App_Path" in your ".gitlab-ci.yml" file to this APP_PATH: '$CI_PROJECT_DIR/'
instead of APP_PATH: '$CI_PROJECT_DIR/src'.
Post committing the above change to Repo this will trigger a deployment job to static webapp (you monitor the job by following the above) Once the job is done when you request the url of the App service you will be able to see the below.
Hope this helps, let me know if you have any further questions on this.