that's very interesting.
But all that you've said is a bit off, for example, you say an ASUS computer will start but not show video, and then you say you tried on two different PC's. This last statement clashes with the first and confuses me.
Also, you say the power button turns the system on, but not off. Are you holding it down to power off or just pressing it once? Because pressing it once to turn it off doesn't work unless you're either still in the BIOS or if you're in Windows and you've set the power button to do that using the power option settings in control panel.
Either way, no video isn't good.
Let's get some more info about your Asus system please.
If it's a pre-built system, please provide the full model number so we can look up the exact specs.
Also, you can provide us with said information using the DirectX Diagnostics tool by pressing the Win+R keys on keyboard, then typing in "dxdiag" without quotations and pressing enter, then looking at the base information from the main tab, and getting graphics information from the display tab.
If it's a custom built machine please tell us all the main components like so:
CPU
Motherboard
RAM
GPU
PSU (Power Supply)
And don't say simple things like 500w PSU and Intel Core i7 but be specific about the models like so:
(I will use my personal specs for this example)
Intel Core i7-6700K
Asus Z170 Pro Gaming/AURA
16GB (2x8GB) 2400MHz Kingston Value RAM
EVGA GTX 1080 Ti SC Black Edition
Corsiar CX600 600w PSU