Rebuilding Hive in Windows XP Pro.

Assuming that System Restore was on, there should be multiple older copies of the System file along with the other registry files.  In my experience you have to replace them all together.

Microsoft has a bulletin on the process:
https://support.microsoft.com/en-us/kb/307545

My preferred method is to boot BartPE (CD with bootable Windows XP) and save some steps:

Boot BartPE
Go to a command prompt
Go to the drive letter of your hard drive (probably C:)
cd \windows\system32\config
md regbu
move system regbu
move security regbu
move default regbu
move sam regbu
move software regbu

Then you use the explorer to find the System Volume Information folder on the boot drive
Look in the _Restore..... folder.  You'll see a number of snapshots.  Go into the most recent and you'll see a folder named Snapshot.  You can copy the five registry files from there to \windows\system32\config.  Then you'll rename each to get rid of the REGISTRY_.... stuff at the beginning.

I should mention that with BartPE I've often found that I can only do one thing with the explorer then it seems to hang.  For example, if you rename the _Registry_machine_system to system it will not let you rename another file.  If so, just open another explorer and do the next file.  Otherwise you can do the rename at the command line.

OK... I should have paid closer attention to your post.  You're on the same link.

The copy that isn't working is just so that you can save what you have (your old registry).  It's a good habit to be in even though you think the registry is bad or is missing a file.  You can skip any of those copies that tell you the file doesn't exist.

If you don't already use BartPE I'd highly recommend looking into it (if you will be working on more XP machines).  It makes this job a lot easier.

As far as copying from a different computer, no, you don't want to.  These files are very hardware- and user-specific.

The registry files are only backed up to c:\windows\repair when you run a system state backup.  You can do a directory listing to see all the files that are present in a folder/directory using the "dir" command.

If you have access to another machine where you can pull the drive from the non-functioning machine and connect it to the functioning one as a secondary drive, the steps can be made a bit easier as you could operate in a GUI environment if you're not used to working at the command prompt.

You can proceed on to part 2 in the link.  But really, the important part is to grab the registry files from a snapshot in SystemVolumeInformation, and copy them (while also renaming them) into c:\windows\system32\config.  And by the way, sometimes it is also necessary to delete the .LOG files in that location.  If I recall correctly, these contain pending changes to the registry, and sometimes they will write a change that makes the machine unbootable again, forcing you to redo all the steps of copying the registry files again.

>>  Can I simply copy and paste this file from a different Xp Pro machine  <<  NO - you would have a backup of that system
when in doubt if the system file exist - use the command dir in the folder where it resides - to show all files in there

but if the file does not exist - you can't use this method

>>> ... this BSoD I've been getting about a missing System File ... Obviously, I am missing the "System" file. <<<

What exactly does the blue screen show?
You refer to "a missing system file".  Is it possible that this is some file other than the file named "system"?

the link shows where the files are :  c:\windows\system32\config\system

is your system set to show hidden files and folders?

If System Restore was on (which is generally likely) then the Microsoft link you found would have done the trick.  Did you ever look to see if those files existed in System Volume Information?  It is fairly quick to identify and to resolve.

The preferred method that I mentioned was to use BartPE, a copy of Windows XP that is bootable from a CD (or USB stick).  You may be able to do this from the Recovery Console, but you may have issues with permissions.

BartPE (or alternatives such as UBCD) is VERY useful for XP repairs and well worth the effort to create it.  I realize that they are getting rarer.

>>  The machine will not boot. How would I access the System Volume Information in that case?    << you can hook the drive to a working system, to see, or copy what you need
also you have a couple more options with darts : https://technet.microsoft.com/en-us/windows/hh826071.aspx?f=255&MSPPError=-2147217396

but as usual with MS it is not free..lol

To be clear.....   if you have done this before, it's about a 10-minute process to resolve (boot BartPE, save whatever there is of the old registry, copy last registry from System Restore/System Volume Information, rename registry files, reboot).

Just as a follow-up, and to serve as clarification for anybody else trying to list system files using the DIR command, you usually need to use the   DIR /AS   switch to show system files, and   DIR /AHS   to  show hidden system files (although just the /A switch usually also works).