  Hundreds of black holes with massive main-sequence companions (OB+BHs) might
be identified from Gaia astrometry with the Astrometric Mass-Ratio Function
(AMRF). We investigate the impact of blue supergiant companions (BSG) instead
of dwarfs and the presence of additional companions in the system that are
unresolved by Gaia on the astrometric identification of OB+BHs. We also explore
how accurate the primary mass needs to be constrained. Moreover, we assess how
the high-precision publishing constraints of astrometric binary orbits in the
latest Gaia data release DR3 impact the detection of OB+BHs. We establish a BSG
mass-magnitude relation and compute BSG AMRF curves. From a mock population of
non- and single-degenerate massive binaries and triples with OB or BSG
primaries, we asses the false-positive identification fraction and the effect
of the BSG AMRF curves. We compare the number of stars with astrometric DR3
orbits in the second Alma Luminous Star catalogue (ALSII) with new predictions
on the OB+BH detection using the conservative DR3 publishing criteria. BSG
primaries and triples are not expected to impact the false-positive
identification fractions significantly. However, if the evolutionary stage of
the primary star is unknown, the usage of the BSG curves is still recommended.
This also reduces the OB+BH identification fraction significantly. The primary
mass does not need to be known to benefit from the high identification fraction
of OB+BHs, while keeping the fraction of false-positives low. We find no OB+BH
candidates in the ALS II among the DR3 orbits. This null-detection cannot be
attributed to the underlying BH-formation scenario, but rather to the stringent
DR3 selection criteria. To infer the BH-formation scenario with Gaia, we
propose that the constraint on the relative parallax precision in DR4 should be
95% less conservative than the DR3 criterion.(Abridged)
