We analysed the stellar population properties of a well-defined sample of 552 galaxies at redshift 0.6<z<0.77 drawn from the LEGA-C spectroscopic survey. This paper is the first of a series, and it is aimed at i) presenting the catalogue of revised absorption indices for LEGA-C DR3 and of the inferred physical parameter estimates while describing their systematic uncertainties and at ii) deriving benchmark scaling relations for the general massive galaxy population at intermediate redshift. We estimated light-weighted mean ages and stellar metallicities through careful analysis of key absorption features in the stellar continuum spectra of the galaxies coupled with photometry. The observables were interpreted in a Bayesian framework with a comprehensive library of model spectra based on stochastic star formation histories, chemical enrichment histories, and dust attenuations. We discuss various sources of systematic uncertainties within our method as well as systematic differences with results from other spectral fitting approaches. We derived volume-weighted scaling relations connecting light-weighted mean ages and stellar metallicities with galaxy stellar mass for the general galaxy population at <z>=0.7 and masses >10^10^M_{sun}_. We find the downsizing trends observed in the local Universe to be already in place 6Gyr ago. We also observe a bimodal distribution of light-weighted ages as a function of mass, transitioning around 10^11^M_{sun}_. Such a bimodality is not observed in the stellar metallicity-mass relation, which changes from a steep to a flat regime across Mstar~10^10.8^M_{sun}_. Similar trends in age and metallicity also emerge as a function of velocity dispersion, but with a sharper transition from young to old around log(sigma_star_)=2.3. Differences with respect to the trends as a function of stellar mass suggest that age is primarily dependent on velocity dispersion below and above the transition regime, while both the stellar mass and the depth of the total gravitational potential well (as traced by the velocity dispersion) contribute to stellar metallicity. We release the catalogues of revised absorption index measurements for LEGA-C DR3 used in this work and of the inferred stellar population physical parameters to public repositories.