Coal and lime probably don’t have a high enough inductance to be heated by induction. Induction furnaces are limited to metals as far as I know. Also, the temperature required for this process is almost 4000°F, which is difficult to obtain by combustion. Thus, electric arc is about the only way to go.
But the process does just require heat, right? So if you had a magical furnace that could reach that temperature, it would work as well? Or does the electric arc itself do anything to the chemistry?
Yes, you could probably use a laser furnace if it was cost-effective at those power levels. But it’s not even close. Arcs are and very practical in almost every sense and ridiculously cheap besides the power they consume. And a laser would consume more.