Did the experiment after visiting daughter who boiled water in electric kettle than transferred into a pan, and I asked why, she said speed, however gas hob rated 5.5 kW kettle rated 2.8 kW so gas hob nearly twice the size, so she did demo.
No stop watch just filled kettle to mark, poured into pan with lid, refilled kettle and switched both on together, and the electric kettle won by a huge margin.
So I returned home, same experiment however this time 3 kW induction hob and 2.8 kW kettle, kettle actually won but by such a small margin a few seconds so could say they were the same.
Clearly a massive amount of heat produced by gas hob goes into the room, plus of course a load of moisture and combustion gases, hence you really need a cooker hood with gas that takes moisture and gases outside, where with electric a carbon filter is good enough.
To calculate the cost you also have to factor in how much energy is used to heat or cool air in the room, I had a non flue type gas heater in a caravan OK for odd cold day but damp soon became a problem, to heat a space you need the heater to draw any combustion air from outside and also send all combustion products outside, this is exactly what the central heating boiler does, so yes filling the pan with hot water may save energy as the central heating boiler does not draw air from the room. With a hot water tank in the loft you needed special mixer taps where the water is only mixed at the spout, but with a combi boiler these are not required so one would assume the water is potable, other wise it would not be permitted to pre-mix water.
However our simple kettle is long gone, the kettle we have today boils a cup full at a time, so filling a pan with boiling water would take a long time, and we use an induction hob so rather pointless.
But be it the tumble drier or the kettle, when one type pumps air outside and other does not be it the cooker extractor or the tumble drier vent time of year and outside temperature will have a massive effect on calculations. Also how room is cooled, in the other house our velux windows work well, no cooker extractor required, just open rook windows, it may cool kitchen but not rest of house, and kitchen does not catch the sun, this house very different, sun shines in through kitchen windows so normally warm, only real way to vent is open back door, which in summer means flies. For some reason flies do not seem to come in to the Velux window, but they do from other windows, maybe they do come in but also find there way out easy?
So it clearly changes summer/winter, and house to house, so it would be very hard to measure. I am sure a wall mounted boiler gas fired with a flue outside would be cheapest to run, and with Quooker and like the idea of piped boiling water is already here, personally don't think their safe in domestic, OK with a work place where everyone is trained, but not domestic.
I think gas is great for central heating where combustion air is drawn from outside and the flue gases are also taken directly outside, for the caravan I accept the gas oven, hob and fridge, you have not always got electric mains power and generators are too noisy, but not seen a gas fridge in a house for years, and I am sure gas cookers now we have induction will slowly disappear, OK some will still want them, some people still use solid fuel cookers, but it will be a minority.
Must admit as a child my mother used solid fuel, kettle on the corner always simmering, but that was when Shotton was a steel working town and the steel works could not use small coke so sold it off very cheap, when steel works closed so solid fuel fires were ripped out. I can walk into the kitchen when adds start on TV put coffee in two cups place them both under the two one cup boilers and return with two cups of coffee before the adds finish, even if I need to top up boilers with water, it would not matter how cheap, if I can't make coffee during the adds, I don't want it.
