I read that appliance at certain times in the U.S. (for example stoves) were grounded through the neutral wire. Since the neutral wire carries current does this means the case of the stove itself is carrying current. Why didn't homeowners get electrocuted with such a setup? Is it because as long as the neutral is connected/grounded properly the current is going to "chose" to take that path (least resistance) instead of the homeowner (much resistance)?
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.
The body of the appliance isn't "carrying" current, because current flows between two points of differing voltage. A grounded object, however, is held to the same voltage potential of whatever it is connected to. In simpler terms, the only way for current to flow OUT of the body of an appliance (say the frame of a stove) is for current to flow IN to the frame of the stove from some other higher potential source. Since (hopefully) the stove isn't wired to anything "hot" (i.e. at high potential relative to ground) then no current will flow through the frame. Think of birds on a power line...although they are connected to a high potential, they aren't connected to anything else, so no current flows. In the stove example, the frame is connected to neutral (which is tied to ground at the panel), and is held at 0V. Unless something touches the stove that is at a higher potential--be it a live wire or a person wearing a sweater--no current flows. In the latter case, only a transient current flows until the person reaches the same potential as the stove.
It's not a matter of "path of least resistance", so much as it is that current flowing into the stove from the neutral wire would have nowhere else to go. The stove is no different from a bird on a wire, and just as no current flows through the birds (regardless of how much is flowing through the wire itself), no current flows through the stove.
The reason the grounding wire is now separate is that--in some cases--the neutral wire can end up at a different potential from ground (i.e. there is a voltage difference between the neutral wire and a person standing on the ground). In that instance, current will flow from high potential to low potential (from the charged neutral line to the person touching the stove) the same way it would if the person had touched any other live conductor at high potential. In modern day wiring, the separate grounding wire will preferentially "divert" any stray currents away from users via the "path of lowest resistance" effect.
It takes some puzzling over to get the hang of it, but hopefully I haven't confused the issue further!