Does it make sense to switch to single chainring on an older mountain bike?
The more general form of the questions is: is it worth upgrading a bike vs buying a whole new bike?

In many cases it is not worth it. Individual upgraded components are compromised by remaining lower end ones. Full upgrading of components is as expensive as a new bike. The bike is still saddled with older technology (such as rim brakes).

However, if the point is to mess around wrenching on a bike, give it a new lease of life, the components to be replaced are worn out anyway, and the limitations are well understood; then that in fine. Especially so if the builder is prepared to spend some time on Craigslist and Ebay finding cheap serviceable parts. 

[Full disclosure - I own a mountain bike with components spread over two decades.]