Lets try this can you press and hold the '''T''' key when you restart your system. Basically we want to see if the system will go into Target Disk Mode.
If it does go into TDM let it run for awhile (an half hour is more than enough). Try going into Diagnostics this time holding the '''D''' key.
Let us know what you find!
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=== Update (07/07/2019) ===
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We its starting to should like your logic board has failed here. Here’s the needed replacement board you’ll need [https://www.thebookyard.com/product.php?cPath=96_218&products_id=6746|Mac Mini Logic Board - 2.4GHz - GeForce 320M (2010) - Recycled] and here’s the guide to put it in [guide|3115]
Lets try this can you press and hold the '''T''' key when you restart your system. Basically we want to see if the system will go into Target Disk Mode.
Reference: [https://support.apple.com/en-us/HT201255|Mac startup key combinations]
If it does go into TDM let it run for awhile (an half hour is more than enough). Try going into Diagnostics this time holding the '''D''' key.
Let us know what you find!