Mac Mini A1347 (2014) Boot and Video Issues. Possible faulty logic board

The two best options are to try running the Apple Diagnostics.

Second option would be to try for an Option-Boot to see if you can get the Apple boot picker menu and let the computer sit there. If it can sit here (even if no boot devices available), it will at least tell you the power supply is working well enough to sit in this limited state. If no boot volume is shown, then your internal drive is probably bad and the source of the problem.

Unfortunately there is no good way of determining if the power supply or Logic Board is at fault without trying another power supply. You could, however, try plugging the Mini into a different electrical outlet (or directly if using a power/surge strip).

While our organization does not have that many Mac Minis, I rarely recall any issues with either the Logic Board or even power supplies in any model of the Minis. The internal drive is about the only part I replaced regularly.

Have you tried booting into recovery? I would try that (hold down Command-R when powering up). If that fails then there is a major malfunction that could likely be logic board related or possibly power supply.

The only way to do that is to take it to an Authorized Apple Service Provider and have them check it out and they can give you the results. Who knows, it could be something as simple as a power switch, though doubtful that that is the case.

Also, being a 9 year old computer, it may be time to move on to something newer.