Would you use a cauldron in your kitchen today? Probably not. A pot is more practical for use on your kitchen stove.

Cauldrons fell out of favour when the open wood fire fell out of favour.

Britain saw shortages of wood starting in the sixteenth century, particularly around large cities. This was a catalyst for increasing use of enclosed stoves.

Enclosed stoves burn wood more efficiently and are safer and more comfortable to use. They are also more suitable for the burning of coal—on the rise due to wood scarcity.

Here is an example of a kitchen at Hampton Court. It is in a bit of a transitional state. The fire is semi‐enclosed and cauldrons on legs have been employed. It would not be hard to install grates and use flat‐bottomed pots instead.

The transition took centuries and happened sooner in some places than in others, but as the open fire’s popularity declined, so did the cauldron’s.

A large cast iron pot would continue to be used for various outdoor tasks well into the 19th c. A common one would be laundry: clothes would often be stirred in an iron pot over a fire ( yes, not much fun..ask someone at living history museum like Sturbridge Village what their least favorite job is, and laundry is very high up on the list) . Another would be for scalding slaughtered pigs or chickens, to make it possible to get the hair/feathers, and that would be something seen well into the 20th c., especially in the Appalachians after the first frost, in "hog-killin' time".

To keep the round bottom pots from rolling over in the fire, they typically had three stubs on the bottom. Sometimes you will find them with a raised rim a little ways from the base- that would allow them to fit into a tripod base, or ( for smaller pots) actually set into an open hole in the top of a woodstove. There actually would be small, low two-holed woodstoves meant for doing laundry.

For boiling water for cooking, brewing etc. copper was better- iron tended to taint the flavor. In the earlier 19th c. , it would be a copper kettle ( sometimes mistakenly called a teapot now) that would be the hot water heater for the house, and a large farm might have a "brewing copper" for making beer, which could also be used for making jellies, preserves.

As to why you don't find many "cauldrons" in antique shops, a possible explanation is that they're cast iron, and brittle. Leave one outside in the winter to fill with water and freezing can crack it. Drop it while shifting around, and the same can happen, maybe even knocking out a chunk. Cast iron is hard to weld or patch today, practically impossible in the days before arc welders. But those broken pots are still put to use. If you're vigilant you will still see them utilized in garden landscapes, turned so that the crack or missing section is hidden.

Lydia Maria Child: The Frugal Housewife

John Fuller : The Art of Coppersmithing