Thanks harun. I did reboot the Raspberry 3 a few times and reinstalled Hyperion and suddenly it all worked in Kodi now.
Probably will still do the colour calibration and then connect the USB grabber + HDMI splitter etc to get it working from any HDMI source but I have gotten much further now.
So just to answer my own question. Kodi at least via OSMC should work outright when everything is set-up correctly. No additional software / plugin needed. I believe the internal grabber needs to be enabled however for Kodi to work (please correct me if I am wrong)
2nd observation: I used the colour picker and the colours were all wrong when leaving it at RGB. Red was blue and it was all over the place. Had to change it to BGR for me (I tried one after the other) and that displayed the colours correctly according to the colour wheel.
This may help someone in the future.
For me the current issues are solved. I will see how it goes when adding the USB Grabber / HDMI splitter etc.
Thanks again.