We actually use fire, hot gas, for welding about as often as ordinary welding with an arc.
Three broad families can be selected: chemical energy to hot gas - torch, arc to hot gas - plasma cutter, arc to metal - welding.
Why is it that ordinary welding is seen by you more often? Probably because your interest lies in joining pieces of metal, not separating. Both torch and plasma cutter blow fast stream of gas, that blows molten metal away from the joint. It is possible to weld using them, but required control of the jet of hot gas is much more precise than with an arc.
Arc allows to transfer the heat to the molten metal without blowing it away, and leaving a void.
As to your question about temperature and power, it is best estimated using black body radiation, power and area
If temperature is above 200 C you can ignore all other types of loss of heat in normal conditions, like gas convection. And at 2000 C heat loss from the radiation dominates so much that only boiling can match it.
In welding temperature is so hot that it produces UV, temperature is about 10 000 C of the arc itself. So the radiation dominates, and everything else can be ignored if max temp is of interest. Plasma cutter and torch can reach 3000 C, but it is still enough for most metals. So temperature is not an issue, you can use any of them to melt steel.
It is the control that people want. And arc provides this control. And ease of use. No more chaotic stream of gas nearby, so the welding is easier.
About your idea of using resistance, keep in mind that there is a few volts drop just because electrons leave or enter a metal. And that is close to what arc welding has in the first place. So even a perfectly conducting plasma will still heat the metal a lot, because of the energy of the electron that is spent by the power supply and liberated in a form of heat as electron travels the boundary between metal and non-metal.
And thats why welding polarity matters. it says on which side will electron produce this heating.