Benefits of a total refurb
 

So currently own a 1993 Trek Singletrack 903 MTB. The frame and most of the parts are in good shape, the components are XTR (a few years old) and the only real thing that needs replacing is the shifters and chain rings. From years of use it just has been worn and no matter what there is dirt everywhere. Not dirt from a normal ride but the kind that sticks there and is more of a permanent muck.

Would it be benefitial for me to take apart the bike piece by piece and thoroughly clean and re-grease each part, or would that cause more problems by risking an substandard rebuild or breaking a part?

I rebuild all my bikes once a year.
If done properly with the correct tools a rebuild will insure the longevety of your bike.
There will be some cost involved in replacing worn parts. (cables, cogs, sealed bearings)
Most used bikes I have worked on that were scrapped were a result of a minor maintenance issue that could have been easily avoided with preventative maintenance.

Enjoy

If your bike is 13 years old and has not had the bearings cleaned and repacked, it's long overdue.

A good set of brushes will let you clean the bike without disassembly.