If you want to manually install server-side Google Tag Manager on App Engine in the Google Cloud Platform, you are of course welcome to do so. The main benefits of doing this are that you get to choose the region where App Engine is deployed and that you can use existing project resources instead of having a new project automatically created for you. However, to run the script without errors, there are a number of things you need to enable first.
The automatic provisioning process of the Google Tag Manager server-side tagging service is extremely useful. With just a few clicks of the button, you can have a fully functional (albeit limited to testing use) server-side tagging endpoint on the Google Cloud Platform (GCP). For a video overview of the automatic deployment, see this video. However, the automatic provisioning process creates a new Google Cloud Platform project and, at the time of writing, always deploys the App Engine application (on which the tagging server runs) in the us-central-1 GCP region.
Google Tag Manager is strictly a tag delivery system, and it’s very careful not to collect any analytics data on its own. This is most likely a deliberate choice, because if GTM was to start collecting data, it would introduce additional barriers to adoption. Nevertheless, being a tool that consolidates the design, development, deployment, and testing of all the marketing and analytics pixels, code snippets, and utilities running on a website or a mobile app, lacking the necessary features for auditing and monitoring has always seemed like an oversight.