Correction -
That said, isn't the pH of water considered to be chemically neutral even if its pH is either higher than any acid or lower than any alkali?
No, the pH of water is 7 (if pure). The pH of even dilute H2SO4 is 1 to 2- 7 is higher than 1, the pH of water is much higher than that of sulfuric acid.
The pH of something like household ammonia is on the order of 10-11 or more, higher than that of water.
pH is the negative log of the H+ concentration, and it works out that the equalibrium concentration of pure water is 1x10^-7, and the negative log of that is 7.