We investigated the factors that critically affect the half-metallicity of the quaternary Heusler alloy $\mathrm{C}{\mathrm{o}}_{2}(\mathrm{Mn},\mathrm{Fe})\mathrm{Si}$ (CMFS) by examining the film composition dependence of the saturation magnetization per formula unit, ${\ensuremath{\mu}}_{s}$, of CMFS thin films and the tunneling magnetoresistance (TMR) ratio of CMFS/MgO/CMFS magnetic tunnel junctions (MTJs). We also investigated the origin of the giant TMR ratio of up to 2610% at 4.2 K (429% at 290 K) obtained for CMFS MTJs with Mn-rich, lightly Fe-doped CMFS electrodes. Co antisites at the nominal Mn/Fe sites ($\mathrm{C}{\mathrm{o}}_{\mathrm{Mn}/\mathrm{Fe}}$ antisites) can consistently explain the ${\ensuremath{\mu}}_{s}$ for (Mn + Fe)-deficient CMFS thin films being lower than the half-metallic ${Z}_{\mathrm{t}}\ensuremath{-}24$ value and the TMR ratio for MTJs with (Mn + Fe)-deficient CMFS electrodes being lower than that for MTJs with (Mn + Fe)-rich CMFS electrodes. It was revealed that the $\mathrm{C}{\mathrm{o}}_{\mathrm{Mn}/\mathrm{Fe}}$ antisite is detrimental to the half-metallicity of the CMFS quaternary alloy, as it is in the ${\mathrm{Co}}_{2}\mathrm{MnSi}$ (CMS) ternary alloy. It was also shown that $(\mathrm{Mn}+\mathrm{Fe})$-rich compositions are critical to suppressing these harmful antisites and to retaining the half-metallic electronic state. In addition, a relatively small Fe ratio, rather than a large one, in the total $(\mathrm{Mn}+\mathrm{Fe})$ composition led to a more complete half-metallic electronic state. Half-metallicity was more strongly enhanced by increasing the Mn composition in Mn-rich, lightly Fe-doped CMFS than in Mn-rich CMS. This phenomenon is the cause of the giant TMR ratio recently reported for CMFS MTJs. Our findings indicate that the approach to controlling off-stoichiometry and film composition is promising for fully utilizing the half-metallicity of quaternary CMFS thin films as spin source materials.