Why is HCl the acid of choice for swimming pools?
There are several reasons not to consider these acids to be interchangeable for use in swimming pools. A total answer covers aspects of chemistry, biology, safety, availability, and cost. As hydrochloric, nitric, and sulfuric acids are all strong acids and can provide the desired acidification, they are very different when you consider their respective counter ions. Do you want chloride, nitrate, or sulfate in your pool water? 

Safety First:


Hydrochloric, nitric, and sulfuric acid are all strong acids that are capable of causing chemical burns and present inhalation hazard from their fumes. 
Additional safety information can be obtained from PubChem.: 
Hydrochloric acid 
Sulfuric acid 
Nitric acid 


Chemistry:


Nitric and sulfuric acids are more corrosive to copper and stainless steel. Copper (I) chloride is significantly less soluble in water than copper salts of nitric and sulfuric acids. Swimming pool heaters are likely to contain copper pipe. 
As an added benefit for salt pools, which use an electrochemical chlorine generator, the Cl- ions are converted to chlorine for disinfecting the pool water. 


Biology:


As pointed out in another response, nitrates can promote algae growth. 


Availability and Cost:


Hydrochloric acid is commonly available as muriatic acid (31.5% HCl, ~10 molar) from home centers, building and construction suppliers, and hardware stores, because it is used in many construction-related activities, with cost ranging from \$5 - 9 (USD) per US gallon. A gallon will contain approximately 37 mole equivalents of H+. 
Sulfuric acid is available from auto parts retailers as battery acid (37% H2SO4) and costs approximately \$10 (USD) per US quart ($40 per gallon). A gallon of battery acid will contain approximately 36 mole equivalents of H+ (adjusting for normality). Sulfuric acid costs 4-8 times more than hydrochloric acid for equivalent pH reducing capacity. 
Nitric acid is less readily available to the consumer market. One on-line supplier offered reagent grade (70%) for \$125 (USD) per liter. 


Added Note:

Cyanuric acid is a weak acid and serves as a stabilizer. Quoting from PubChem:


  Cyanuric acid is used in swimming pools to lower the rate of photochemical reduction of chlorine, hypochlorous acid and hypochlorite ion.