imac startup failure - hear startup sound, get apple logo, then shuts down after ~5 seconds

Here's how this got resolved. Had to use a USB keyboard because wireless was not responding. Fortunately, most important data was already in icloud - verified that first, so when target mode didn't work, went to reinstall the OS. First aid on MacOS base system had no effect. Reinstall of Catalina failed. No reason given but my speculation is a corrupted disk partition, only 17 GB of free space. Tried to reinstall Yosemite (original OS), that failed - no disk found. Ran disk utility, it reported that the 128 GB SSD was no longer fusion. So I erased the SSD. This time reinstall of Yosemite found the HDD and was successful. Reconnect of wireless keyboard also successful At that point I basically had the original machine in original condition, no data or apps other than the basic content (welcome to 2014!). HDD now shows 1 TB free. Downloaded Catalina, this time install successful. So after about 5 hours of effort, back in business. Not sure how much I trust the HDD in the machine, but otherwise it's still a nice computer, 21" display is perfect for current uses. Thought I was headed to the recycling center, glad that wasn't necessary.

I also called Apple Tech support, suggestion was to try to try to recover data. I am able to put the iMac in Target disk mode, and connect another Mac via Thunderbolt 2 cable. Alas, the other Mac in disk utilities does not recognized the HDD in the iMac. Any last thoughts/tricks to try to get the HDD in the target iMac to be recognized?

(Hard to get to an AASP when most places are closed due to COVID).