The decline of speech intelligibility in presbycusis can be regarded as resulting from the combined contribution of two main groups of factors: (1) audibility-related factors and (2) age-related factors. In particular, there is now an abundant scientific literature on the crucial role of suprathreshold auditory abilities and cognitive functions, which have been found to decline with age even in the absence of audiometric hearing loss. However, researchers investigating the direct effect of aging in presbycusis have to deal with the methodological issue that age and peripheral hearing loss covary to a large extent. In the present study, we analyzed a dataset of consonant-identification scores measured in quiet and in noise for a large cohort (n = 459, age = 42-92) of hearing-impaired (HI) and normal-hearing (NH) listeners. HI listeners were provided with a frequency-dependent amplification adjusted to their audiometric profile. Their scores in the two conditions were predicted from their pure-tone average (PTA) and age, as well as from their Extended Speech Intelligibility Index (ESII), a measure of the impact of audibility loss on speech intelligibility. We relied on a causal-inference approach combined with Bayesian modeling to disentangle the direct causal effects of age and audibility on intelligibility from the indirect effect of age on hearing loss. The analysis revealed that the direct effect of PTA on HI intelligibility scores was 5 times higher than the effect of age. This overwhelming effect of PTA was not due to a residual audibility loss despite amplification, as confirmed by a ESII-based model. More plausibly, the marginal role of age could be a consequence of the relatively little cognitively-demanding task used in this study. Furthermore, the amount of variance in intelligibility scores was smaller for NH than HI listeners, even after accounting for age and audibility, reflecting the presence of additional suprathreshold deficits in the latter group. Although the non-sense-syllable materials and the particular amplification settings used in this study potentially restrict the generalization of the findings, we think that these promising results call for a wider use of causal-inference analysis in audiology, e.g., as a way to disentangle the influence of the various cognitive factors and suprathreshold deficits associated to presbycusis.