About 20% of all B-type stars are classical Be stars. The Be phenomenon is strongly correlated with rapid rotation, the origin of which remains unclear. It may be rooted in single- or binary-star evolution. In the framework of the binary channel, the initially more massive star transfers mass and angular momentum to the original secondary, which becomes a Be star. The system then evolves into a Be binary with a post-main-sequence companion, which may later be disrupted in a supernova event. Hence, if the binary channel dominates the formation of Be stars, one may expect a strong lack of close Be binaries with main sequence (MS) companions. Through an extensive, star-by-star review of the literature of a magnitude-limited sample of Galactic early-type Be stars, we investigate whether Be binaries with MS companions are known to exist. Our sample is constructed from the BeSS database and cross-matched with all available literature on the individual stars. Out of an initial list of 505 Be stars, we compile a final sample of 287 Galactic Be stars earlier than B1.5 with V<=12mag. Out of those, 13 objects were reported as Be binaries with known post-MS companions and 11 as binaries with unknown, uncertain or debated companions. We find no confirmed reports of Be binaries with MS companions. For the remaining 263 targets, no significant reports of multiplicity exist in the literature, implying that they are either Be binaries with faint companions, or truly single. The clear lack of reported MS companions to Be stars, which stands in contrast to the high number of detected B+B MS binaries, strongly supports the hypothesis that early-type Be stars are binary interaction products that spun up after mass and angular momentum transfer from a companion star. Taken at face value, our results may suggest that a large majority of the early-type Be stars have formed through binary mass-transfer.