OS : Live USB bootable of Linuxmint-21.1-xfce-64bit on a flash stick, 4GB USB2.

Hardware A:
MB : G45T AM2 V:1.0
CPU: Intel Core2 Quad 2.40 GHz
RAM: DDR2 800 SDRAM 2G
GPU: On board Intel® G45
BIOS:American Megatrends V.02.16

Hardware B:
1: PSU Corsair RM650x
2: PSU TrendSonic 450w

Issues:
Hardware A + B1.
Ran very well for a month. Then, tried to install Xfce onto a hard drive, Seagate Momentus 5400.4 120G. The system would not start after adding the hard drive to the to it. Disconnected the hard drive. The system would not start either or sometimes only the CPU fan ran for about 0.5 second, more than a twitch.

Hardware A + B2.
Had the hard drive disconnected. Replaced Corsair RM650x with TrendSonic 450w, and power on, the CPU fan spun continuously, but no beep, all mouse, keyboard and monitor were not activated.

All the trouble was caused by connecting a hard drive to the hardware system. And the trouble did not disappear when the hardware system restored its previous status.

Your suggestions, please?

Thanks.

Was this a known good hard drive or did it come out of another broken system?

Thank you.

"Test HDD "Seagate Momentus 5400.4 120G" on another system."
The HDD is a good one. It was used and formatted on another PC about a minute prior to connecting to this PC.

"Remove CMOS battery,"
Did this but did not make any difference. Also, the CR2032 was replaced with a relatively new one from another PC. Did not help.

Thank you.

"Was this a known good hard drive ...?
It was a good one used on another PC for storing backup files.

This looks like a "switch to death". It's common that hardware components are killed by switching power on due to start-up current peaks. It may only be a memory module or a PSU but in many cases the mainboard itself is killed.

My thoughts, the psu doesn’t have enough wattage. Old psu 650, current psu 450, something went wrong when connecting the hard drive, it was just coincidence and it was time the motherboard died.

I would autopsy the PSUs if they're out of warranty. Look for electrolytic capacitors whose tops are no longer perfectly flat, or have obviously leaked anything. A failing or failed PSU can kill a motherboard.

If the motherboard has electrolytics of 470uF or more anywhere near RAM or CPU socket, inspect them too. If any exist branded OST, suspect them as well, as they can fail without any outward clue.

Thank you.

"the mainboard itself is killed."
The chips in southbridge? How could it be so bad? This means the whole thing, the MB is useless if so.

Thank you.

"it was time the motherboard died."
The whole setup is a waste if so. The MB was manufactured in the year of 2007 or 2008. 14 to 15 years ago. It is very difficult to find one now. Or, the cost would be sky high if anyone could be found on the online market.

"Time to say goodbye …"

It might be cheaper to buy a more or less complete computer of this era than a seperate mainboard. Ask your friends. Some people plan to dispose their computers of this era. You have a memory module, a HDD and a PSU if replacements are required.

On the other hand, think about new hardware. An inexpensive e.g. Ryzen 5 based system will offer much more performance than a Qore2 Quad based system. Ancient ports like PATA (aka IDE) or Parallelport (aka LPT) can be provided with PCIe x1 cards if they are required.

Thank you.

Autopsy the PSUs.
Beyond my ability. Yes, I took a few PSUs apart to clean then in the past, but not anything beyond cleaning.

"if they're out of warranty."
Corsair RM650x is out of warrenty.
TrendSonic 450w is a brand new one but of a inferior brand. I bought it at the time just for testing.

"capacitors"
Checked them on the MB with a flashlight and did not find popped tops or swelling bellies.

"OST"
Don't know if there are capacitors of that brand because I can't see all the brand names or marks on the capacitors.

Thank you.

The motherboard, CPU, RAM and other basic parts are all about 14 to 15 years old. I am not trying to save them for use but to monkey around with them for fun and for learning Linux. I am going to get rid of Windows a year or so later though I am now using Windows 10.

This is my second try. The first one, every part is the same except for the hard drive and monitor, was completed successfully a month ago.

Another thought, did you recheck and make sure everything is seated and plugged in correctly cables, memory, cards

Thank you.

Yes, checked and rechecked many times and even reassembled the whole setup many times.