Resource not accessible when trying to read/write CheckRun

Relates to “Resource not accessible by integration” when requesting GitHub Pages build via REST API

I build a GitHub App. I ensure my App request properly check:write, and this has been granted by my own organisation to given app. However, when I try to read CheckRuns/Create a new CheckRun, I encounter 403: Resource not accessible by integration. I indeed rely on a server-to-server token, as the point is to create this checkrun on most push events.

What may I be doing wrong?