The promise by the LIGO/Virgo/Kagra (LVK) collaboration to detect black hole-neutron star (BH-NS) mergers via gravitational wave (GW) emission has recently been fulfilled with the detection of GW200105 and GW200115. Mergers of BH-NS binaries are particularly exciting for their multi-messenger potential, since the GW detection can be followed by an electromagnetic (EM) counterpart (kilonova, gamma-ray burst, afterglow) that can reveal important information on the equation of state (EOS) of NSs and the nature of the BH spin. This can happen whenever the NS does not directly plunge into the BH, but rather is tidally disrupted leaving behind debris to accrete. We carry out a statistical study of the binary stars that evolve to form a BH-NS binary and compute the rate of merger events that can be followed by an EM counterpart. We find that $\gtrsim 50\%$ of the mergers can lead to an EM counterpart only in the case BHs are born highly spinning, otherwise this fraction never exceeds about $30\%$ for stiff NS EOSs and a few percent for soft NS EOSs. However, the possibilities that BHs are born with near-maximal spins and that NS internal structure is described by a stiff EOS are disfavored by current LVK constraints. Considering that these values only represent an upper limit to observe an EM counterpart due to current observational limitations, as in brightness sensitivity and sky localization, BH-NS mergers are unlikely multi-messenger sources.