This estimate has been made Dec. 11, 1752 [1] and has been summarized [2] as follows:

There seems to be a calculation error in the above statement for the fire engine: Drawn water in 24 hours = volume/stroke ⋅ strokes/minute ∗ 60 minutes ∗ 24 hours ∗ duty cycle =(0.3048/2)² ∗ π ∗ 1.524m ∗ 8 ∗ 60 ∗ 24 ∗ (18/24) = 961 m³ = 211,390 imperial gallons. If we convert to US gallons we acutally arrive at a number of  253,869, which is a deviation of 1.3% from the reported value. If I would be able to see the original calcuation maybe this mystery could be solved. 

Another irregularity seems to be the amount of water drawn in 24 hours by horses. Two horses worked in a team for three hours, which constituted a shift. Thus a 24-hour working day would be divided into 8 shifts. If in one shift the horses draw 35 tubbs * 80 gallons of water = 2,800 gallons, 8 shifts would draw 8* 2.800 = 22,400 gallons of water. This is a third of the 67,200 gallons cited. By calculating the horsepower for 67,200 gallons drawn in 24 hours we would arrive at 2,800 gallons of water = 12,730kg raised by 73.152m = 12,730kg∗73.152m*9.80665kg/s2 = 9,132kNm per hour = 2.54 kNm/s = 3.4 horsepower for a team of two horses. This is far too much if the horses work continuously over a longer time. It is, therefore, far more reasonable than the 2 horses produced 1.13 horsepower as a team of two or 0.57 horsepower each. It is known that Watt generously rounded his definition of horse power upwards, so that his customers could not accuse him that his machine performed less powerful than promoted. 

If we compare the gallons of water pumped up in 24 hours by the fire engine = 250.560 with the 8*2800gal. for the horses we arrive at a proportion of 1⁄11 and not 1⁄3 as suggusted in above comparison.

Apart from the calculation errors indicated above, the calculation omits the Newcomen engine's running costs, particularly the coal costs. As the Newcomen engines were running for decades the erection costs must be stretched over the expected lifetime. Of course, in the beginning no one could have an idea how long these machines would be running. Assuming a lifetime of 20 years these makes 20*365 days/year = 7300 working days. I am not sure if the pumps had to work even on Sundays, the miners usually worked 6 days per week, with only a few extra holidays. The erecting costs of £365 mentioned above are then £1⁄20 = 1sh. (This is just a rule of thumb to get an idea of the figure.)

As it is, this source is unhelpful to decide whether a Newcomen fire engine would provide the same power at less costs. Another issue that remains to be investigated is if the horse droppings were environmental more friendly than the smoke produced by steam engines.