Why didn't homeowners get electrocuted when appliances were grounded through the neutral?
The US distribution system uses a neutral conductor which is grounded at the service entrance of each house/building. The neutral conductor and the ground conductor are physically and electrically bonded at the main service entrance, and tied to actual earth ground via rods and/or metallic water pipes as well.  

The devices which were commonly "grounded via neutral" were exclusively 240V primary loads. In the US system those loads are fed by two hot conductors, each 120V away (in opposing directions) from ground/neutral. No current flows in the neutral conductor from one of these loads (generally, they are not even connected to it.)  If there was any 120V power used in these items, it was a minor load such as a clock motor. The neutral conductor, however, was full-sized for rated load of the appliance - 30A for a typical dryer, 50 A for a typical stove. So any 120V load that would actually cause current to flow in the neutral conductor would cause a small current to flow in a large wire that was grounded at the far end. As such, no appreciable voltage would ever arise on the neutral conductor.

The move to a 4-wire system with separate ground conductor is, notably, not one where the powers that be decreed that all your old devices needed to be ripped out and replaced since they were scary and hazardous to own; new devices need to meet the new standard. How much "real improvement in safety" there may be is, frankly, debatable; but that debate is somewhat pointless (4-wire is a done deal) and probably also off-topic. 

In general, even a current-carrying neutral on a fully-loaded 120V circuit should not have sufficient voltage above ground to cause a shock. Correctly sized wires should limit the total voltage drop in the wiring to 3%, or 3.6 volts, 1/2 of which will lower the hot voltage and 1/2 of which (1.8 volts) will appear as a difference between neutral and ground.