Electric drip coffee makers heat the water a little bit at a time. The heating makes steam bubbles that lift the water and a check valve causes steam pressure impulses to push the water to raise the water from the tank at a slow speed. This is similar to the pumping in a coffee percolator.
To use a bulk of water heated on a cook top you will need an elevated tank with a flow control device, maybe just a small orifice. The tank might need insulation.
Replacing the drip coffee maker heater with a propane powered heater to work the same way requires a thermostat to turn off the gas when the water runs out and it overheats.
As soon as the first bit of hot water hits the grounds the brewing begins. If you have to heat all the water first the time to have coffee takes longer. Overlapping the heating time with the brewing time is the clever part.
What I have found that works for me is the aero press in the inverted mode. There is very little clean up and the taste is good. Heating and brewing do not overlap in time but you only make one cup at a time. Brewing a second cup overlaps drinking the first.