Aperture photometry uncertainties
Because the uncertainties in the number of counts detected in each pixel are considered to be independent. That means some of them are higher than the "true value" (the count rate you would measure if you observed for an infinitely long time), whilst others are lower. To some extent these independent uncertainties will cancel, and the net result is that for normally distributed errors the correct protocol is what you have described.

Perhaps a good way of seeing this is that suppose you took 100 independent measurements of the same thing, each with its own, roughly equal uncertainty. If I asked what the uncertainty in the average was, you wouldn't just add up all the errors and divide by 100 because that would give an uncertainty that was identical to that of an individual measurement. Instead you would do the quadrature sum of the uncertainties and divide by 100, which reduces the uncertainty in the average by a factor of $\sqrt{100}$.