If Jesus performed the cleansing of the sanctuary in the first century, what about the scapegoat ritual?

The book of Hebrews shows that Jesus did in fact make purification for sins in the first century, and did fulfill the type of the entry of the high priest with blood. Jesus fulfilled all the blood ministration at His ascension.

However, there is another portion of the Day of Atonement that is not spelled out in Hebrews--the portion of the service dealing with the scapegoat.

Since the scapegoat is not spelled out in any text of the New Testament that I am aware of I am hesitant to be dogmatic on the question.  When the New Testament relates something we can confidently accept it. When it does not then we can certainly make some inference from the type. However, as we have already seen when looking at the fulfillment that Hebrews does spell out we may not always get the full picture just by looking at the earthly type. The type is a shadow, and not the true reality. With that in mind, here are a few thoughts:

The scapegoat portion of the service happened after the high priest left the sanctuary, having offered the blood of the Lord's goat for cleansing.  Therefore it may well be that the scapegoat is not spelled out in Hebrews because it had not yet happened, and will not until Jesus leaves the heavenly sanctuary for His second coming. The burden of the author was to point out the superiority of Jesus to the Old Testament rites. He showed this through what Jesus had already accomplished at that time.

If the scapegoat does not occur until Jesus leaves the heavenly sanctuary,  it will likely take place at the end (just as the Day of Atonement is near the end of the calendar year for the Israelites), and may involve the final removal of sin.

The cleansing portion of the Day of Atonement happened in the sanctuary at Jesus' ascension because of the once-for-all nature of Jesus' sacrifice, entry, and blood presentation. It had to happen at one point in time.  It was never to be repeated. However, the rest of the Day of Atonement may well wind up corresponding to its relative time in the Hebrew calendar, just as the other holy times appear to show correspondence.

Beyond the timing of the fulfillment, it should also be noted that the Lord's goat and the scapegoat are both animals that are without blemish, and suitable to be used as a sin offering. In fact the two together are called a sin offering, and the only distinguishing factor is the where the lot falls.

This makes it likely that it simply represents another aspect of Jesus' ministry, as did the rest of the sanctuary service.  This is especially true when we realize that the scapegoat was used to make atonement:

Lev 16:10  but the goat on which the lot fell for Azazel shall be presented alive before the LORD to make atonement over it, that it may be sent away into the wilderness to Azazel.

For this reason I tend to think that the scapegoat ritual prefigures in some way Jesus' final removal of sin from the world and His people in the future. However, since the New Testament does not spell out the details, we will just have to wait to see.