Backpropagation is a specific algorithm for computing gradients of composite functions, but even the failures that do come from composition (multiple sequential sigmoids cause exponential gradient decay) are not backpropagation specific: that's just how the gradients behave for that function, whatever algorithm you use. The remedy, of having people calculate their own backwards pass, is useful because people are _calculating their own derivatives_ for the functions, and get a chance to notice the exponents creeping in. Ask me how I know ;)
[1] Gradients being zero would not be a problem with a global optimization algorithm (which we don't use because they are impractical in high dimensions). Gradients getting very small might be dealt with by with tools like line search (if they are small in all directions) or approximate newton methods (if small in some directions but not others). Not saying those are better solutions in this context, just that optimization(+modeling) are the actually hard parts, not the way gradients are calculated.