That exact statement, "stabilize voltage to earth during normal operation", dates all the way back to telegraphy days. It made more sense then and was of greater importance, especially when trying to troubleshoot a bad circuit. This statement existed when the system included the source. That is, the supply was not independent, but a component of the entire system. Today, the utility service provider grounds their systems, so we must to be default on the premise wiring side since there will ALWAYS be a potential (path back to the source) from any ungrounded conductor to earth on a modern day system.
Think of it this way. The source limits and regulates the voltage between any two ungrounded conductors. On a grounded system (one conductor intentionally connected to earth), the voltage between any ungrounded conductor and any non-current carrying metal part properly bonded and/or connected via a EGC, OR the earth, OR a body that serves in place of the earth, the voltage will be the same at any point (save for voltage drop). However, on a system where no connection is made to earth at ANY point (even at the source), the voltage between the ungrounded conductors and metal parts, the earht, etc. can be from zero to infinite. This would be considered unstable.