The coil in the fire would work just fine to heat the tub - although it could be more efficient. You could capture better heat just by enclosing the coil in bricks or something, decreasing the amount of fuel needed to heat the tub.
To my eye it works on the same principle as many wood-burning stoves that have water jackets installed, and which can provide plenty of hot water to even a sizeable home.
As an example, a few of my neighbours heated all their water all year long with just their wood stoves. When the fire was going, the heated water rose all the way to the reservoir on or near the roof. Just by turning the tap and letting gravity do the rest, this hot water could then be used in the kitchen, for showers, etc. Good ol' thermo-siphons.
The presenter in the video below starts talking about how this works at 1:34:19. There are other points in the vid that may be interesting, such as a serpentine coil system attached to the stove, which starts at 39:40. There are various other methods discussed.
For thermo-siphon systems, it is very important to have pressure release valves at the high point of the system and on top of the storage tank to let steam or oxygen bubbles out - otherwise things start blowing up!
Also, if such a pot has a lid and is not too big, one might easily carry the hot water pots to the kitchen, bathroom or laundry area, keeping them in (DIY) insulated boxes.
FWIW, I found this tutorial about a cheap DIY hand-washing 'machine' using 5 gallon buckets and a plunger. The music is pretty obnoxious but the concept is simple enough, and would be cheap and easy to make. I was thinking it'd probably be much more effective in cold times to heat water and use some 'machine' like this rather than anything else. 5 gallon buckets make an easy toilet as well... just gotta make not to mix up the toilet bucket and the laundry one...