Technically you would be correct in the UK
To be fair, Under IEC / BS7671 standards the neutral conductor in the same Edison circuit if replicated outside the US would be classified as a PEN conductor, Protective Earth Neutral, and thus you would be correct in calling it a combined neutral-ground.
The NEC however requires it be treated it as a grounded conductor, that only by special exceptions given in the code, can ground the frame of the range or dryer.
In Eastern Europe it was common to have TN-C up to the sockets. You would only have two wires, Live and PEN, and the PEN conductor would land on the earth terminal and then jumper to the neutral terminal. Today this is forbidden, a hold over practice from the days of the soviet union.
TN-C earthing system: a distribution system in which one live part of a power source is earthed, exposed-conductive-parts of an electrical installationare connected to the earthed live part of the power source by PEN, PEM or PEL conductors.
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You are correct, you'd need at least 2 failures, and with the recent code changes you'd need 3 failures. Code now requires most US dryers to be GFCI protected. The idea is that if you lost the equipment grounding conductor with a simultaneous insulation break down the GFCI would act as a last resort and trip.
Which receptacles in the laundry room require GFCI protection?
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I think, that as time goes on all circuits in the US will require some form GFCI or GFP protection regardless of voltage, wiring method, rating, or equipment installed. Equipment grounding conductors can and do become compromised in a world where newer standards of safety dictate that multiple things must go wrong first before any danger becomes present.