One can indeed derive the Born Rule as a natural consequence of the direct-action picture of fields (in which both the emission of a quantum and the absorption of that quantum are required for what counts as 'measurement'). This is shown explicitly here: https://arxiv.org/abs/1711.04501To put it less complicated. The justification for Born's rule simply is that it works. Today, there's seems to be no way to derive Born's rule from the other postulates of QT, and that's why it's taken as an independent postulate..
In a nutshell: you must multiply the amplitudes for emission and absorption to get the amplitude for the entire process (which is the only way that radiative processes occur in the direct action theory), and of course that total amplitude turns out to be the Born probability. If one only considers part of the process (either emission OR absorption), then that gives you the probability for either one. The reason this is not derivable in the 'standard' QT approach is because it tacitly assumes that emission OR absorption occur unilaterally, and thus neglects the additional complex conjugate factor, so one has to put it in as an ad hoc postulate.