We develop a sophisticated model of FRB observations, accounting for the intrinsic cosmological gas distribution and host galaxy contributions, and give the most detailed account yet of observational biases due to burst width, dispersion measure, and the exact telescope beamshape. Our results offer a significant increase in both accuracy and precision beyond those previously obtained. Using results from ASKAP and Parkes, we present our best-fit FRB population parameters in a companion paper. Here, we consider in detail the expected and fitted distributions in redshift, dispersion measure, and signal-to-noise. We estimate that the unlocalised ASKAP FRBs arise from $z<0.5$, with between a third and a half within $z<0.1$. Our predicted source-counts (logN--logS) distribution confirms previous indications of a steepening index near the Parkes detection threshold of $1$,Jy,ms. We find no evidence for a minimum FRB energy, and rule out $E_{rm min} > 10^{38.5}$,erg at 90% C.L. Importantly, we find that above a certain DM, observational biases cause the Macquart (DM--z) relation to become inverted, implying that the highest-DM events detected in the unlocalised Parkes and ASKAP samples are unlikely to be the most distant. We do not expect our quantitative estimates in this region to be accurate until it is directly probed with localised FRBs. Since the cause of this effect is a well-understood observational bias however, it is guaranteed to be present to some degree. Works assuming a 1--1 DM--z relation may therefore derive erroneous results.
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