Context. The initial distribution of rotational velocities of stars is still poorly known, and how the stellar spin evolves from birth to the various end points of stellar evolution is an actively debated topic. Binary interactions are often invoked to explain the existence of extremely fast-rotating stars (vsin i ≳ 200 km s−1). The primary mechanisms through which binaries can spin up stars are tidal interactions, mass transfer, and possibly mergers. However, fast rotation could also be primordial, that is, a result of the star formation process. To evaluate these scenarios, we investigated in detail the evolution of three known fast-rotating stars in short-period spectroscopic and eclipsing binaries, namely HD 25631, HD 191495, and HD 46485, with primaries of masses of 7, 15, and 24 M⊙, respectively, with companions of ∼1 M⊙ and orbital periods of less than 7 days. These systems belong to a recently identified class of binaries with extreme mass ratios, whose evolutionary origin is still poorly understood. Aims. We evaluated in detail three scenarios that could explain the fast rotation observed in these binaries: it could be primordial, a product of mass transfer, or the result of a merger within an originally triple system. We also discuss the future evolution of these systems to shed light on the impact of fast rotation on binary products. Methods. We computed grids of single and binary MESA models varying tidal forces and initial binary architectures to investigate the evolution and reproduce observational properties of these systems. When considering the triple scenario, we determined the region of parameter space compatible with the observed binaries and used a publicly available machine-learning model to determine the dynamical stability of the triple system. Results. We find that, because of the extreme mass-ratio between binary components, tides have a limited impact, regardless of the prescription used, and that the observed short orbital periods are at odds with post-mass-transfer scenarios. We also find that the overwhelming majority of triple systems compatible with the observed binaries are dynamically unstable and would be disrupted within years of formation, forcing a hypothetical merger to happen so close to a zero-age main-sequence that it could be considered part of the star formation process. Conclusions. The most likely scenario to form such young, rapidly rotating, and short-period binaries is primordial rotation, implying that the observed binaries are pre-interaction ones. Our simulations further indicate that such systems will subsequently go through a common envelope and likely merge. These binaries show that the initial spin distribution of massive stars can have a wide range of rotational velocities.