It is applied through faith. No, mankind does not make it efficacious, it is efficacious regardless of belief or unbelief.
efficacious - adjective
capable of having the desired result or effect;
He blood is quite capable. In fact, it is available to those who would receive it.
2Ti 3:15 "...from childhood you have been acquainted with the sacred writings, which are able to make you wise for salvation through faith in Christ Jesus."
Jam 1:21 Therefore put away all filthiness and rampant wickedness and receive with meekness the implanted word, which is able to save your souls.
One has to look no further than the parable of the sower to get a grasp on this 'implanting' of the word. And one has to look no further than the parable of the sower to see the 'lack of regard' of the implanting of the word.
Man cannot be his own savior. But God has given ALL mankind the choice to make. We cannot say that just because some choose not to accept the sacrifice, then it is not effective. It is quite effective, and available, for those who would receive it. What we have done is misunderstood what the sacrifice was for. We think that it is some kind of 'magical' wash that makes us god. It does not. What exactly does the sacrifice of Christ do?
Hbr 9:22-28 Indeed, under the law almost everything is purified with blood, and without the shedding of blood there is no forgiveness of sins. Thus it was necessary for the copies of the heavenly things to be purified with these rites, but the heavenly things themselves with better sacrifices than these. For Christ has entered, not into holy places made with hands, which are copies of the true things, but into heaven itself, now to appear in the presence of God on our behalf. Nor was it to offer himself repeatedly, as the high priest enters the holy places every year with blood not his own, for then he would have had to suffer repeatedly since the foundation of the world. But as it is, he has appeared once for all at the end of the ages to put away sin by the sacrifice of himself. And just as it is appointed for man to die once, and after that comes judgment, so Christ, having been offered once to bear the sins of many, will appear a second time, not to deal with sin but to save those who are eagerly waiting for him.
His sacrifice was to 'put away' sin. To purify us from sin. Is that the 'magical' potion that guarantees us a place in the Kingdom?
Hbr 9:11-14 "But when Christ appeared as a high priest of the good things that have come, then through the greater and more perfect tent ( not made with hands, that is, not of this creation) he entered once for all into the holy places, not by means of the blood of goats and calves but by means of his own blood, thus securing an eternal redemption. For if the blood of goats and bulls, and the sprinkling of defiled persons with the ashes of a heifer, sanctify for the purification of the flesh, how much more will the blood of Christ, who through the eternal Spirit offered himself without blemish to God, purify our conscience from dead works to serve the living God.
The sacrifice, as was intended for in the Old Covenant, was to give the one sacrificing 'peace' with God. That way this thing called 'missing the mark' did not weigh on their conscience and cause them to keep stumbling along saying 'woe is me'. But now, we have been given an ETERNAL redemption, one that we did not have any part in, it is a sacrifice made without hands, and therefore is perfect in every way. It now continually cleanses the mind and soul of the one who believes. But if a person does not believe, then it has no effect on them.
It cleanses us to enable us to serve the living God. Its not a 'magical' wash that secures us a place in the Kingdom. What it does is secure the ETERNAL redemption of the conscience of those who believe. That is why, when He returns, it is not to deal with sin, but to take back those who are eagerly waiting for Him; those who through a pure conscience are serving Him.
What one has to understand is what this 'redemption' is all about.
redemption - lytrōsis
1) a ransoming, redemption
2) deliverance, esp. from the penalty of sin
What do we understand about this 'ransom' that was paid? This 'deliverance' from sin?
Rom 6:16-22 "Do you not know that if you present yourselves to anyone as obedient slaves, you are slaves of the one whom you obey, either of sin, which leads to death, or of obedience, which leads to righteousness? But thanks be to God, that you who were once slaves of sin have become obedient from the heart to the standard of teaching to which you were committed, and, having been set free from sin, have become slaves of righteousness. I am speaking in human terms, because of your natural limitations. For just as you once presented your members as slaves to impurity and to lawlessness leading to more lawlessness, so now present your members as slaves to righteousness leading to sanctification. For when you were slaves of sin, you were free in regard to righteousness. But what fruit were you getting at that time from the things of which you are now ashamed? For the end of those things is death. But now that you have been set free from sin and have become slaves of God, the fruit you get leads to sanctification and its end, eternal life."
See, we are confined in our choices. We either can choose to obey, or to disobey which is called sin. When we obey from the heart, the Blood of Christ cleanses our concision to serve the Living God. In that serving, we are sanctified, through that sanctification we will become Christ like, and in the end we will receive eternal life.
All mans role is to do is obey. Thats it.