I am looking for (finding or writing) a script that can record a video and after that playback it over HDMI. And if possible also delete it after the playback.
It's unlikely you will find an existing script to do exactly that. But off the top of my head it would not be too hard to write one.
It's trivial to initiate video recording in a script.
Liikewise switching to playback mode after you stop recording.
Getting the script to play the most recent video (assuming HDMI is correctly configured) likely requires faking button presses but that's easy too. It can be tricky to get the sequence robust however.
The file i/o functions available to scripts allow deletion although you'll need a bit of logic to figure out the right filename to remove.
Finally a scripted reboot will probably be needed to sync the Canon firmware to the modified file system.