I probably ought to leave this alone but there are few subleties here that create the sort of misunderstandings that drive a physics teacher insane
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The load resistance is irrelevant as power consumption in various parts of the circuit is at issue and the load resistance is doing the useful part of the energy expended. So the design of the load is simply a matter of matching it to the supply to get the desired job done.
The key is to figure out the power used in the rest of the circuit as that is not doing the desired work. Efficiency is the ratio of the power doing what you want the power to do compared to the power that's doing other things that can't be avoided.
And, yes, squares do get involved because of the available design parameters for the power distribution system. This is related to why circuit breakers are rated in amps yet your interest is in the power going to your load. The primarly limiting factor in electrical system power distribution is resistance. The relationship between power and resistance does involve a power of two exponent. Electrical power systems are fixed voltage, variable current where the wire size, which determines resistance losses in the circuit, is the primary cost factor. (note that similar concerns influence circuit design for the pins on the Picaxe.)
Yes, you can select a voltage for an electrical power delivery system to use but changing that voltage has its own losses. This is why AC is used for the grid, why cars are heading from 12v to 42v systems, why computers often run off 12v or 48v DC to move the reduction to volt levels needed by processors as local as possible, why household (US, anyway) use split phase systems, why many industrial services use 480 v 3 phase, and so on. Modern electronics is helping with the efficiencies of voltage conversion as you can see if you look at the 5 and 3 volt regulator options for our Picaxe boards. Heaters only need a resistance element to match desired power level to supply voltage. Most other appliances and devices have a much more complicated situation and have to do things like voltage changing, current limiting, and power conditioning to match the device to the available power source.
If you get into line losses for meter reading concerns, you are in the ballpark where you have to consider how much of the line between the meter and the heater is in the desired heating space (its heat losses are helping the heater!), the amount that the heater load resistance changes with temperature as it heats up, losses incurred in regulating temperature, connection losses, - good brainstorming project to make a list of these things. Don't forget metrology concerns, either.