We present the initial results of an ongoing survey with the Karl G. Jansky Very Large Array targeting the CO(J = 1-0) transition in a sample of 30 submillimeter-selected, dusty star-forming galaxies (SFGs) at z = 2-5 with existing mid-J CO detections from the Atacama Large Millimeter/submillimeter Array and NOrthern Extended Millimeter Array, of which 17 have been fully observed. We detect CO(1-0) emission in 11 targets, along with three tentative (∼1.5σ-2σ) detections; three galaxies are undetected. Our results yield total molecular gas masses of 6-23 × 10¹⁰ (α _CO/1) M _⊙, with gas mass fractions, f _gas = M _mol/(M _*+M _mol), of 0.1-0.8 and a median depletion time of (140 ± 70) Myr. We find median CO excitation ratios of r ₃₁ = 0.75 ± 0.39 and r ₄₁ = 0.63 ± 0.44, with significant scatter. We find no significant correlation between the excitation ratio and a number of key parameters such as redshift, CO(1-0) line width, or Σ_SFR. We only find a tentative positive correlation between r ₄₁ and the star-forming efficiency, but we are limited by our small sample size. Finally, we compare our results to predictions from the SHARK semi-analytical model, finding a good agreement between the molecular gas masses, depletion times, and gas fractions of our sources and their SHARK counterparts. Our results highlight the heterogeneous nature of the most massive SFGs at high redshift, and the importance of CO(1-0) observations to robustly constrain their total molecular gas content and interstellar medium properties.