I took it to several shops, but no one was willing to work on it... in one case, because they had a service agreement for Google devices, and the Pixel C (along with the Pixelbook) were devices they were not allowed to do any hardware service for.
However, I have it working again, for now at least, after I took a look at the teardown and then pressed pretty hard on the screen at the place where the ribbon cable connects (center right). Here's hoping that fix lasts a while!