For some strange reason, the computer gets hung and reset itself. After reset, booting was not possible because no O.S. was detected.
This is a classic indication that you have a storage device that is about to fail.
I used Hirens Boot CD USB to boot Hirens tools, and it did. It recognized the HDD, and when I performed a scan, it threw that the first sector is bad.
The fact Hirens detected a bad sector on the drive is direct evidence the problem is hardware related.
I know that UEFI does not have MBR so I think it could be worse if I try to repair that sector.
This is false. Every UEFI implementation I am aware of supports BIOS emulation, which allows you to use MBR, and I am unaware of any major OEM manufacturer who has implemented UEFI firmware that doesn't support this legacy BIOS emulation.
Unfortunately, the Hirens Boot CD USB does not boot in the damaged PC when the HDD is connected. After boot, it immediately shows a message telling that the HDD has problems. I cannot even load BIOS.
On the UEFI machine, if you enable this emulation mode, you can use Hirens. Your inability to run it currently is due to the fact, Hirens, does not support UEFI.
Is there a way to repair that in a non-UEFI PC?
This entirely depends on how many bad sectors. If the data is true that valuable professional data recovery services should be used, every failed attempt you make will make data recovery more unlikely.
If I connect other HDD (the HDD that belongs to the old PC), it does not boot either, but I can load BIOS and Hirens Boot CD from the USB drive.
Your current machine is configured to only allow EFI compatible environments. You would have to enable the BIOS emulation mode for you to have any chance of booting from the old HDD. However, that isn't likely to work, due to the massive differences in the hardware between the two machines.
Any help to try to repair this disk, please?
If the HDD has too many bad sectors, there is no repair that kind of damage. The best you can hope for is to extract the data from the failed HDD. However, every attempt to access a file on a bad sector could make the problem worse. If you are hoping to boot from this failed HDD, you are simply out of luck.
The best you can hope for is, when you connect the HDD that does not boot to the machine that does, you can view and access the files you want. You really should create an image of the HDD before you do this; however, in my personal experience, just reading from the device can make the problem worse.