It is not what is in Scripture that supports this beyond the command of Jesus Christ to do it, and do it often, the assurance that it forgives sin. What does support our position this the complete absence of, or indication of Apostolic Succession as defined by Rome.
The Small Catechism distills and explains Scripture very simply. Article VI, The Sacrament of the Altar:
VI. The Sacrament of the Altar
As the head of the family should teach it in a simple way to his household.
What is the Sacrament of the Altar?
It is the true body and blood of our Lord Jesus Christ, under the bread and wine, for us Christians to eat and to drink, instituted by Christ Himself.
Where is this written?
The holy Evangelists, Matthew, Mark, Luke, and St. Paul, write thus:
Our Lord Jesus Christ, the same night in which He was betrayed, took bread: and when He had given thanks, He brake it, and gave it to His disciples, and said, Take, eat; this is My body, which is given for you. This do in remembrance of Me.
After the same manner also He took the cup, when He had supped, gave thanks, and gave it to them, saying, Take, drink ye all of it. This cup is the new testament in My blood, which is shed for you for the remission of sins. This do ye, as oft as ye drink it, in remembrance of Me.
What is the benefit of such eating and drinking?
That is shown us in these words: Given, and shed for you, for the remission of sins; namely, that in the Sacrament forgiveness of sins, life, and salvation are given us through these words. For where there is forgiveness of sins, there is also life and salvation.
How can bodily eating and drinking do such great things?
It is not the eating and drinking, indeed, that does them, but the words which stand here, namely: Given, and shed for you, for the remission of sins. Which words are, beside the bodily eating and drinking, as the chief thing in the Sacrament; and he that believes these words has what they say and express, namely, the forgiveness of sins.
Who, then, receives such Sacrament worthily?
Fasting and bodily preparation is, indeed, a fine outward training; but he is truly worthy and well prepared who has faith in these words: Given, and shed for you, for the remission of sins.
But he that does not believe these words, or doubts, is unworthy and unfit; for the words For you require altogether believing hearts.
For many years, and still in some places, the Catholic Church still withholds the precious blood from it's membership; this is another non-biblical practice just like AS.
Tradition is fine as far as it goes, but where it falls down is when it is at odds with Scripture. When it is something that is man made like AS or follows human logic such that if the bread is Christ's body, the consuming it alone will give one both His body and His blood, what ever reasons are given, they do not stand up to Scripture.
In the CC, Tradition often trumps Scripture; in Lutheranism, Tradition must pass the test of Scripture: if it is neither conflicts with Scripture, nor is prohibited by Scripture, there is nothing wrong with retaining many of our traditions. If it is at odds with Scripture, or prohibited by Scripture; out it goes. This traditions which have been retained are one of the reasons Lutheranism often appears to the outsider to be as "Catholic" or more "Catholic" than the Catholic Church. I hope this helps.
Mark