Snub nose bike saddles
There are several companies making "noseless" saddles as well as snub nosed saddles.  Either will "work", as in, you can use them to ride a bike.  I had several riding friends who swore by the Spongey Wonder in particular.  I tried it and found that while it was possible to use, it wasn't for me.  These seats are sold with the idea that they don't contact the perineum and won't do damage to nerves or vasculature there.

A nose breaking on a saddle is fairly rare.  I have seen far more broken rails or shells than I have ever seen broken noses.  As mentioned in comments, I'd likely seek a warranty replacement if this was on a new saddle.  

As for needing the presence of a nose, strictly speaking, it is not necessary to ride a bike.  However, it is an extra control surface that can be used in various positions to exert or maintain control of a bicycle.  That being said, I always prefer to ride a saddle that DOES have a nose.  

TL;DR  If you have nerve or vascular concerns and have had issues with nosed seats, a snub nose or noseless saddle may be worth investigating.  If you are simply concerned with the idea of a nose breaking again, I would say it's a low probability event and not worth investigating alternate saddle designs over.