Bizarre behavior of older MacBook Pro -- auto-restarts, auto shut downs. Diagnosis?

Having extreme problems with an older MacBook Pro (6,2 -- mid-2010). Worked perfectly fine until a week ago when it suddenly would (at first) just turn "off" randomly -- not a formal shut down, but the screen would just instantly go black, and then it would just turn off, as if it was a desktop computer that had been unplugged. Pressing the power button would then cause it to start up again, reloading the OS from scratch. Strangely, though, it never displayed a "This Mac was shut down improperly" error warning, as what would happen if this was a kernel-panic freeze.

This issue then escalated until it didn't just "turn off" randomly, but would also sometimes restart itself randomly -- even when not being used and just sitting there, the screen would again instantly turn black and then the restart chimes would sound, and it would restart from scratch, even though I hadn't even touched the computer.

This escalated even further to the point where it would only be able remain "on" for 15 minutes at most, before either turning off or restarting without explanation or cause. This occurred whether or not it was plugged in, whether or not I was using it, whatever the battery charge was, and would even happen if I had put it to "Sleep." The length of time it remained on was random, not always the same number of minutes.

Further escalation as of two days ago led to it not reliably or consistently turning back on when the power button was pressed. Sometimes it would take multiple attempts at pressing the power button before finally starting up.

Thinking it might be a problem with the System or the software., I ran "Disk Utility" several times to "Repair Permissions," which the program seemed to do successfully. Whenever I ran Disk Utility, it always seemed to find new problems to fix, but always did so apparently successfully. But even after multiple Disk Utility fixes, the same problem persisted.

Thinking the issue might be a dead battery, I just replaced the old worn-out battery with a brand new one, installed it successfully, and turned the computer on -- and everything was fine, the new battery was accepted, recognized and functioning -- until after just 5 minutes the computer once again just went black and turned itself off. But now -- it won't turn on, even though I have now attempted to press the power button over 20 separate times.

So, what is causing this behavior? If it's not a battery issue, and not a software/system issue, then what is it?

Could this be the cause?: Some time ago, the connector cable for one of the two cooling fans inside the case got torn, and possibly no longer supplies power to one of the fans. Could this one issue with the motherboard cause the entire computer to cease functioning?

Any ideas, clues, suggestions or recommendations greatly appreciated. I will answer any questions if needed for clarification.