The ring system , and fused plugs, means you don't need individual circuits for each socket, and because each ring is fed via two pieces of 2.5mm cable it's effectively 5.0mm wiring - yet uses much less cable (and copper) than radial wiring.
The fault modes stated on that page are mainly to do with using spurs which is frowned upon in the UK (may even be disallowed). The "loose connection" fault is an installation error and, as noted on that page, would result in a fire in a spur system.
In Australia, where I now live, they use the spur system and electrical fires seem far more frequent here than the UK.
Edit, one advantage of the ring system is that it's almost impossible to loose the earth connection. Not as important now but before ELCBs, very important. Especially at 230V rather than the finger tingling 110V.
What is the immediate fire risk you are talking of?
If a wire breaks in a spur system any down stream outlets would not get power not burst into flames.