Aspire A515-51-3509 System Starts to Boot then Shuts Off, Cycle Repeats Indefinitely

I've searched through web and forum posts for about two weeks. I've found quite a few things that I've tried and none of them are really pointing me in a useful or definitive direction. I'll try to describe the problem, maybe I'm missing something.

The problem:

- Press power button, blue LED comes on, CPU fan spins up, HD spins up, I can hear and feel the HD accessing/reading the discs - takes about 4-5 seconds for this to happen
- I hear and feel a slight sound from the HD and the whole system shuts off. 
- About 5 seconds later the cycle repeats, seemingly forever.
- This happens on power and with a new battery. Nothing appears on the screen at any point.

Recent activity:

My daughter says she's had trouble with the display going out during use. She just wiggles the screen a little and it comes back. I figured there must be a loose cable since I installed a new LCD about a year ago. She says this boot cycle thing has been happening for about a year, slowly getting worse. Most recently, she's left it in this boot cycle for two days and it eventually booted up.

What I've tried:

I opened it up and one of the video cable ends, the one on the back of the LCD, was loose. I plugged it in completely and try to boot again with no change. 

I unplugged the video cable at the motherboard and plug in an external monitor, with no change.

I've discharged capacitors and unplugged peripherals one at a time, RAM, keyboard, touchpad, BIOS battery, with no change. I don't have another RAM card to try so I've only removed it and booted without it. There's 4GB on board so that should be a sufficient test unless the problem is the onboard RAM, which I can't isolate. 

With a multimeter I tested the battery. It had less between 2 and 5 volts on various pins and appeared to be slightly swollen/inflated so I replaced it with a new one. New battery charged completely (LED went from orange to blue overnight), with no change. 

I pulled the HD (Seagate 1TB) and used a USB 3.0 to SATA plug-in adapter on my desktop. The drive works fine, I can access it, open files, write and copy files to and from. 

I opened my older HP DV7 laptop I have that is working fine with its' original 250GB drive and swapped the HD's, just to see what happens. I know it shouldn't just boot up but I've got to try something different. The HP DV7 BIOS sees the 1TB drive and it goes into the boot sequence but hits a CHKDSK stop every time. I let it go through a complete test the first time, took 3 hours, and it said the drive was fine but wouldn't boot up completely, as expected. I put the 1TB drive in the second HD slot and returned the old 250GB drive to the primary drive slot. On boot the system wants to check the 1TB drive every time. I skip the check and the system boots up fine and I can access the 1TB drive without issue. So, now I'm thinking boot sector is bad or something similar. First possible progress?

To rule out the boot sector idea, maybe, I put the 250GB drive in the A515 machine and it does exactly the same thing the 1TB drive does; spins up, shuts down. So, back to square one.

Bad motherboard? I don't know what else to check at this point.

Don