All saved in either Covenant would be due to the Cross of Christ, and saved by Faith alone, through grace alone, but the 2 Covenants are not the same, as there is some discontinuity between the 2 of them, not as much as Dispensational see it, but not as little as Presbyterians would see it.I think I can generally agree with those statements yes, but none of that takes away from the NC being the consummation (It is finished) of the overall CoG.
Read the Matthew Henry commentary. It helped me at least. OT (elect) and NT (elect) have always been saved the same way in my opinion, by faith in Christ alone and by the grace of God alone (redemption applied by the HS in OT and NT)..that is the formula of the CoG..made most fully clear in the NC. If the OT saints only needed to do law (which was impossible since the fall on their own), but needed not redemption by grace alone, then none of our dead OT brothers/sisters could have ever been saved....less you believe there is more than 1 way of salvation after the fall. Job/David/ Moses did not enter the CoG after Christ stated "it is finished"...no they were apart of the CoG before the 12 apostles (minus Judas) were a part of the CoG.
I hope this helps. As always I am open to corrections.