Hard drives do not consume much power, lower than 20 watts.
It's highly unlikely this is due to your power supply, unless you have a 600W GPU and even then it should at least start.
It does start after you've uncoupled the hard drive? If not, you may have snagged another power cable somewhere. Have you double checked that you turned your PSU back on after installing the hard drive?
If all else fails, you'll just have to remove every additional component, try to get into the BIOS, turn off, install another component and keep going until something breaks.
But if it doesn't start even without your old hard drive, I'll bet you've either forgotten to turn your PSU back on or maybe the little finicky cable for the power switch isn't in properly.
If it does start without your old hard drive, you can pretty much write off that hard drive. Most likely it's causing a power surge or something and it's actually dangerous putting it in your PC.
Yes, it starts and runs normally if I unplug the hard drive. If I plug any of my conventional hard drives back in, it goes right back to not starting at all. This is true of all three of my hard drives, which were all functioning normally in my prior computer, and they all work normally when they're in a powered USB enclosure.
In that case it just sounds like your on board SATA controller is not working. If you're dead set on using these drives, you could look at a PCIe to SATA board. I have one in my server which has 10 SATA ports.
Would a bad SATA controller stop the entire computer from powering on at all if a SATA drive was installed?
Typically no since the southbridge isn’t entirely necessary. But if something is shorted that shouldn’t be shorted it could make the entire system freak out.
Have you checked the cables/ports on both ends? What about the HDD plugged into power but not data?
Last time I ran it, I had the power and SATA cable connected. I should try with just the power cable--no SATA--and see if it still fails to boot.
Buildapc