If one of the neutrals failed, that would cause the relay to not energize, and thus everything would stay isolated.
That is only true if the direct neutral on that cord failed, but not for infrastructure neutral failures in certain infrastructures - you would still have a path back to the transformer, although it would be ugly. Assuming the grounds of both circuits were both connected at such a device (as the schematic shows), your normal path of current return for the second plug would be:
Plug 2 neutral - receptacle 2 neutral - pedestal 2 neutral bus - feeder 2 neutral - service panel neutral bus - transformer neutral
Now, if the feeder 2 neutral suffers a failure, you get this:
Plug 2 neutral - receptacle 2 neutral - (pedestal 2 neutral bus - neutral/ground bonding screw - pedestal 2 ground bus) - Plug 2 ground - easy220 chassis/ground - Plug 1 ground - (pedestal 1 ground bus - neutral/ground bonding screw - pedestal 1 neutral bus) - feeder 1 neutral - service panel neutral bus - transformer neutral
That is the lowest impedance path. There's also another path here:
Plug 2 neutral - receptacle 2 neutral - (pedestal 2 neutral bus - neutral/ground bonding screw - pedestal 2 ground bus) - pedestal 2 grounding electrode - earth - service panel grounding electrode - (service panel ground bus - neutral/ground bonding screw - service panel neutral bus) - transformer neutral
In both these cases, these paths will allow for reasonable current draw, so your relays will activate and the device will technically *work*, but you may have some odd effects. In the first path you may be overloading pedestal 1's neutral. The second path can create shocks if a human creates a third path to ground through his body -- remember that electricity never takes just one path, it will divide current among multiple paths inversely proportional to the resistance of paths.
(There are other paths as well, through other pedestals, but I think we've established there's enough ways to get return current delivered at this point.
)
A couple of notes:
1. Anywhere you see a (neutral - bonding - ground) or vice-versa in the diagrams above, it's likely that a shortcut was taken and neutrals/grounds were on the same bus, but I spell it out anyway.
2. Any other loads in pedestal 2 (for example, another RV using another receptacle) will happily also use this return path to neutral for 120V loads, meaning you can end up with a heckuva lot more current through the device's ground than you might think.
3. The good news: NEC 2008 changed the requirement for subpanels to use 4-wire feeders to avoid specifically this condition (except detached structures / pedestals with no other metallic paths interconnecting them), and NEC 2011 eliminated that exclusion. So anything built starting with NEC 2011 is immune to this (in a neutral feeder failure, it would just stop working). The bad news: anything built before 2012ish is still susceptible, as standard practice was to use 3-wire feeders because of the cost involved with pulling a fourth wire for ground; and some jurisdictions haven't even adopted NEC 2011 yet.
Now, one solution is to leave one ground floating so that a circuit isn't completed but that is even more dangerous. Let's say that you decided to use plug 2's ground and leave plug 1's unconnected to avoid this problem. If feeder 2's neutral has failed, you've now left the chassis of the device at high potential. This is because the hot is now connected to the chassis:
plug 2 hot - plug 2 relay coil - plug 2 neutral - pedestal 2 neutral/ground bonding - plug 2 ground - device chassis/ground
In this case, the relay will not close because you likely won't get enough current through it, unless it sits in a puddle. However, anyone who touches the device chassis or one of its screws and has a ground connection (feet on concrete, touching another grounded device, etc.) will conduct the plug 2 relay coil current to ground, completing the circuit. This might be enough current to kill a human.
I tried to explain it as simply as I could, sorry for the complexity here.
This is the difference between something working, and something working safely (especially under failure conditions). How often does a feeder neutral fail? More often then you think, especially in campgrounds.