One of the most surprising gamma-ray burst (GRB) features discovered with the Swift X-ray telescope (XRT) is a plateau phase in the early X-ray afterglow light curves. These plateaus are observed in the majority of long GRBs, while their incidence in short GRBs (SGRBs) is still uncertain due to their fainter X-ray afterglow luminosity with respect to long GRBs. An accurate estimate of the fraction of SGRBs with plateaus is of utmost relevance given the implications that the plateau may have for our understanding of the jet structure and possibly of the nature of the binary neutron star (BNS) merger remnant. This work presents the results of an extensive data analysis of the largest and most up-to-date sample of SGRBs observed with the XRT, and for which the redshift has been measured. We find a plateau incidence of 18-37<!PCT!> in SGRBs, which is a significantly lower fraction than that measured in long GRBs (>50<!PCT!>). Although still debated, the plateau phase could be explained as energy injection from the spin-down power of a newly born magnetized neutron star (NS; magnetar). We show that this scenario can nicely reproduce the observed short GRB (SGRBs) plateaus, while at the same time providing a natural explanation for the different plateau fractions between short and long GRBs. In particular, our findings may imply that only a minority of BNS mergers generating SGRBs leave behind a sufficiently stable or long-lived NS to form a plateau. From the probability distribution of the BNS remnant mass, a fraction 18-37<!PCT!> of short GRB plateaus implies a maximum NS mass in the range $ odot