For example, let's say instead of gradient descent you want to do a Newton descent. Then maybe there's a better way to compute the needed weight updates besides backprop?

IIRC, feedback alignment [1] approximates Gauss-Newton minimization. So there is an easier way, that is potentially biologically more plausible, though not necessarily a better way.

[1] https://www.nature.com/articles/ncomms13276#Sec20