The chain won't snap. Chain wear is expressed as percentage elongation compared to new. The recommended replacement is 0.5% on a cassette drivetrain and abut 0.7% on a singlespeed bike. That means that the wear in the pin is minimum even on a badly worn chain and hardly weakens it at all. It would be impossible to wear a chain so badly that the pins would shear off. Besides, chains break by coming apart at the rivet heads, not the pins shearing. A poor shift can break a brand new chain open just as easily as an old one. The rivet heads do not weaken from wear.
However, that does not mean you should ride the chain forever. Chain life is in association with sprocket and chainring life. If your chain has worn to the point where it's pitch is a mismatch with the sprocket pitch, it starts to damage the sprocket. The most economical way to manage a drivetrain is to replace the chain at the wear limit. This will enable you to get maximum sprocket life.
On some singespeeds you can flip the rear sprocket around and it is as good as new again. It is prudent to do so unless you have money to waste.
