We use our catalogue of structural decomposition measurements for the extended GALEX Arecibo SDSS Survey (xGASS) to study the role of bulges both along and across the galaxy star-forming main sequence (SFMS). We show that the slope in the $sSFR$-$M_{star}$ relation flattens by $sim$0.1 dex per decade in $M_{star}$ when re-normalising $sSFR$ by disc stellar mass instead of total stellar mass. However, recasting the $sSFR$-$M_{star}$ relation into the framework of only disc-specific quantities shows that a residual trend remains against disc stellar mass with equivalent slope and comparable scatter to that of the total galaxy relation. This suggests that the residual declining slope of the SFMS is intrinsic to the disc components of galaxies. We further investigate the distribution of bulge-to-total ratios ($B/T$) as a function of distance from the SFMS ($Delta SFR_{MS}$). At all stellar masses, the average $B/T$ of local galaxies decreases monotonically with increasing $Delta SFR_{MS}$. Contrary to previous works, we find that the upper-envelope of the SFMS is not dominated by objects with a significant bulge component. This rules out a scenario in which, in the local Universe, objects with increased star formation activity are simultaneously experiencing a significant bulge growth. We suggest that much of the discrepancies between different works studying the role of bulges originates from differences in the methodology of structurally decomposing galaxies.
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