Recovery boot impossible / iMac screen goes black

We have an iMac (not the youngest, running Sierra) the screen of which goes black after one or two seconds. This has happened progressively over a few days (black screen at random points in time, then after shorter and shorter time intervals).

We tried recovery boot (CMD+R). That gives us two seconds of screen time, then black screen, then two seconds of screen time again if we touch the keyboard 15 minutes later, then black screen, etc. With this we were able get to "Disk Utilities", and click "Erase Macintosh HD". With this, we were able to navigate to "Reinstall MacOS" and had message "Recovery disk cannot be controlled" (it took a few hours to get through those steps at the available 1s-every-15min pace).

We tried, but unsuccessfully to reboot from the internet with "Option+CMD+R" or "Shift+Option+CMD+R" (with an ethernet connection). In both cases we get the beep, then a grey screen for about a second, then black-out.

We would be grateful for any interpretation and suggestion.