If someone wants to make them work in rural areas like this, I think the necessary ingredients will be:
1. Cheaper batteries. These are likely coming.
2. More energy. A meal might require 1 kWh or more. (Or less — scrambled eggs won’t require much energy at all.) This is solvable with more panels.
3. Copper. The coil itself is a decent sized hunk of copper. I assume this is part of why cheap little portable induction cooktops still cost $50 or more.
4. Power electronics? I’m not an expert, and I have no idea how much of the cost comes from the power electronics, but integrating the battery and the induction heater seems like it should result in a dramatically simpler system than, say, producing AC from a battery and then converting that AC into a form that will power the coil. The current list price of the Impulse Labs stove includes a hilariously high power output, and a stove targeting rural Africa could be 1/5 as powerful and would still be fantastic.
I wouldn’t be surprised if someone could squeeze the cost of a decent battery powered stove down to $200 in a few years if they had appropriate scale.