Is there a reason why in particular it is so popular to use monotone cubic Hermite interpolation compare to say, quadratic? I understand the the order of the accuracy gets better with a higher degree, but quadratic splines seem already good enough with second order globally. I wonder if there are other reasons...
While it is possible to interpolate data exactly with $C^1$ quadratic splines, this does not allow us to interpolate monotone data with monotone $C^1$ quadratic splines.
Consider one of the monotone example data sets given to your previous Question, $(0,0), (1,0), (2,1), (3,1)$. As with the monotone $C^1$ cubic splines, the only possible polynomial portions in the first and last subintervals are constants, so that to be $C^1$ on the middle interval imposes four conditions, the function and its derivative at both endpoints $x=1,2$.
While a (monotone) cubic polynomial can be constructed to meet those four conditions, it is impossible for any quadratic polynomial to have a derivative with two roots.