I think that we are in violent agreement. The logical process could go backwards from how it's written on the page (although it doesn't have to, if sufficient

I think that we are in violent agreement. The logical process could go backwards from how it's written on the page (although it doesn't have to, if sufficient theoretical machinery is in place); but the pedagogical process goes in the direction that it's written on the page. Technically speaking, any class that uses partial-fractions decompositions should prove that they always exist (in which case the procedure that starts by assuming that they exist is justified, and no backwards run is needed), or should finish every such problem by checking that the putative decomposition is valid (the backwards process to which you refer).