But why would you want to use such a flimsy technology, if you have horses?
There is no way an engine like this could outrun a lean and speedy chariot with two to four horses. Also, even a primitive mechanism like this may have encountered fatal problems with dirt, dust and sand. (Mind that steam locomotion was first tried for roads, which was a total failure because of the quality of roads, even in major city centers like London.)
There are lots of applications for heat engines that don't involve mobility: grinding wheat to flour, threshing and winnowing wheat, fulling cloth, sewing cloth together, hammering hot iron or annealed cold bronze, blowing the bellows on a forge or bloomery, sawing wood, turning wood or stone on a lathe, weaving cloth, spinning thread, playing music, crushing ore, grinding a new edge on a knife or axe with a grindstone, turning a potter's wheel, writing, kneading bread, calculating, and, as mentioned, raising water (from a well, from a river, into your fields, into a tower, into a salt pond, or out of a mine or swamp). All of these are things that people were already doing before Heron, in many cases devoting their lives to them, but steam power was not applied to some of them until less than a century ago.
In medieval times we can add turning metal on a lathe and winding crossbows.
Think of the most common German surnames, which are all occupational: Müller (grinding wheat), Schmidt (hammering iron), Schneider (sewing cloth), Fischer (catching fish), Weber (weaving cloth), Meyer (owning land), Wagner (making wagon wheels), Becker (baking bread), Schulz (herding peasants), Hoffmann (organizing the court), Schäfer (herding sheep), Koch (cooking food), Bauer (farming), and Richter (judging). This is a somewhat skewed sampling of occupations of men in Germany at the late-medieval or early-modern time surnames were imposed. If you were a Schmidt or a Weber you could expect to spend decades doing hard, physical labor every day, in a fixed physical location, labor that is now mostly done by steam.
I think Napoleon snickered at the first steam ships. Something about requiring precisely-controlled fire on a warship. But I bet the ancient Greeks would have killed to add a wheel to their quinqueremes.
There is no way an engine like this could outrun a lean and speedy chariot with two to four horses. Also, even a primitive mechanism like this may have encountered fatal problems with dirt, dust and sand. (Mind that steam locomotion was first tried for roads, which was a total failure because of the quality of roads, even in major city centers like London.)