Jesus suffered more than mere crucifixion in his hours upon the cross. Obscured by darkness, Matthew 27:45, Mark 15:33, but if we are granted to see it, he suffered for sins not his own, 1 Corinthians 15:3, 1 Peter 2:24. Then he yielded up his life, no man taking it from him, John 10:18. Only after this, was his blood shed, the lance thrust in, releasing both blood and water as witnessed by John the apostle, John 19:34.
That blood was shed, copiously, indicates a death. And that it was shed at all indicates not a peaceful death but one of suffering, and one of sacrifice. Foreshadowed by a multitude of intricate sacrificial rituals ( see all of Leviticus) each of which convey differing aspects of the sufferings and death of the one sent by God so to offer up himself, this momentous event is fully explained in the doctrine of Christ, Romans 6:17, the gospel, Ephesians 1:13, delivered by the chosen apostles of Jesus Christ whom he authorised to express all the truth of it, Matthew 28:19.
It should be noted that the blood itself (in and of its mere organic content) did not purchase anything it was ‘that of his own
blood’ which was effectual, namely the worth of the life that was given and the worth of the sufferings suffered by the one living :
the church of God which he possessed for himself with that of his own blood.
[Literal from the interlinear Englishman's Greek New Testament (Stephanus Text) Acts 20:28 and expressing the article as the demonstrative pronoun.]
Many precise words, 2 Timothy 1:13, are used to convey the extent of the sufferings and death and bloodshed of Christ : redemption, remission, reconciliation and justification come immediately to mind but there are more. The epistles are for our learning, Romans 15:4, that we might fully appreciate and understand the offering up of Jesus Christ : his body, his blood, his life, his flesh - his self, 1 Peter 2:24.
And all was necessary to satisfy the righteousness of God, to fulfil all law, to resolve sin itself in its coming into the world, Romans 5:12-21,to resolve a failed humanity which rebelled against its creator, to resolve the matter of law itself since humanity is specifically warned not to partake of the tree of knowledge of good and evil, Genesis 2:17, Galatians 2:20, and to resolve all that the first humanity has done in making a world that is contrary to the will and purpose of God who made all things, 2 Corinthians 5:19.
So yes, both the sufferings of Christ and the death of Christ (both of which are witnessed to by the shedding of his blood) are essential ; for thus, in his sufferings, he bore sins and thus, in his death, he resolved momentous issues of righteousness : and both are absolutely needful for us that we ourselves might be pardoned from all confessed sins, 1 John 1:9, that sins might be remitted, Acts 2:38, that we might be dead to our first humanity and alive to Christ under his headship, Romans 7:4, that we might be delivered from law, Romans 7:6, and that we might walk in the Spirit, Romans 8: 1-4, and that we might have an inheritance in the world to come, Hebrews 9 : 15.