Impossible in an electric furnace (which could be powered by renewable sources).

Virgin steel requires the higher temperatures of a coke-fed blast furnace.

It does require a reducing gas such as the CO produced by coke, but an electric arc furnace can of course reach higher temperatures. Iron can be smelted successfully from ore even without a high enough temperature to melt it, as in bloomery smelts, but the quality of the resulting metal is quite uneven.

then again we have hundreds of years of economies of scale in place for that process, and (presumably) far less for repurifying steel. I'm not saying it's not possible, of course it's not, but is it financially viable with current infrastructure? I'm genuinely asking, I don't know anything about this.

I'm no expert either but I think you can use the same process, though most steel recycling is instead done more cheaply done in EAFs.