Context. Eclipsing spectroscopic double-lined binaries are the prime source of precise and accurate measurements of masses and radii of stars. These measurements provide a stringent test for models of stellar evolution that are consistently reported to contain major shortcomings. Aims. The mass discrepancy observed for eclipsing spectroscopic double-lined binaries is one of the manifestations of the shortcomings in stellar evolution models. The problem reflects the inability of the models to accurately predict the effective temperature and surface gravity or luminosity of a star for a given mass. Our ultimate goal is to provide an observational mapping of the mass discrepancy and to propose a recipe for its solution. Methods. We initiated a spectroscopic monitoring campaign of 573 candidate eclipsing binaries classified as such based on their TESS light curves. In this work, we present a sub-sample of 83 systems for which orbital phase-resolved spectroscopy has been obtained and subsequently analysed with the methods of least-squares deconvolution and spectral disentangling. In addition, we employed TESS space-based light curves to provide photometric classification of the systems according to the type of their intrinsic variability. Results. We confirmed 69 systems as being either spectroscopic binaries or higher-order multiple systems. We classified twelve stars as single, and we found two more objects that cannot be decisively classified as intrinsically variable single or binary stars. Moreover, 20 eclipsing binaries were found to contain at least one component that exhibits stellar oscillations. Spectroscopic orbital elements were obtained with the spectral disentangling method and reported for all systems classified as either SB1 or SB2. The sample presented in this work contains both detached and semi-detached systems and covers a range in the effective temperature and mass of the star of Teff ∈ [7000, 30 000] K and M ∈ [1.5, 15] M⊙, respectively. Conclusions. Based on a comparison of our own results with those published in the literature for well-studied systems, we conclude that there is an appreciable capability of the spectral disentangling method to deliver precise and accurate spectroscopic orbital elements from as few as six to eight orbital phase-resolved spectroscopic observations. Orbital solutions obtained this way are accurate enough to deliver age estimates with an accuracy of 10% or better for intermediate-mass F-type stars, an important resource for the calibration of stellar evolution models for future space-based missions, such as PLATO. Finally, despite the small size relative to the 573 systems that we will ultimately monitor spectroscopically, the sample presented in this work is already suitable to kick off observational mapping of the mass discrepancy in eclipsing binaries.