In Catholicism, why should Jesus have been murdered on the Cross instead of just dying of old age? Imagine that Jesus wasn't murdered but just died of old age. Would there be a difference in redemption for those who believed in Him? If so, why did he need to be killed?

To be clear, I'm specifically asking about the difference between Christ being violently murdered and simply dying of old age. What is important about his violent murder that is different than a peaceful death?

I would like answers based on Catholic teaching and doctrine.
Introduction

There are a number of ways to approach this question for example my answer to this question: How is the matter of why the Son of God needed to be sacrificed explained in the Roman Catholic Church? contains some essential points that will be common to all answers from a Catholic standpoint. Some of those points are:

1) God is free.

2) God does whatever he pleases in accordance with his nature.

3) God has infinite solutions to each of what to creatures are problems: “With men this is impossible, but with God all things are possible.”.

4) Faced with the "problem" of man's redemption, God could have chosen to act or not to act. Once he chose to act, he had infinite solutions to the "problem".

5) Once he decreed the work of redemption, he also decreed the manner by which the redemption would be wrought.

6) According to Church teaching grounded in scripture and explained by theology, redemption is by blood.

7) There is meaning behind and fittingness of the manner of redemption, in this case redemption by the blood of the God-made-man.

Answer

This answer takes from The Theology of the Precious Blood by Edwin Kaiser, C.PP.S. | Catholic Culture. The articles lays out some key points:

1) That God was not obliged to accept any atonement; God could have condoned the sin and demanded no atonement whatever; God might have been satisfied with partial atonement and granted partial condonation. In all of this God was free, nor would either His justice or His mercy have been impaired by His choice. But God decided that man should be shown mercy and be pardoned.

2) The very act of God assuming created nature [not necesarily human], God abasing himself, would have been more than sufficient to redeem because every act of God-made-creature has infinite value.

Therefore according to Catholic teaching and theology, Christ need not have been "violently murdered" and could have "simply died old age", and this could have been more than sufficient for the redemption.

Mary could have brought forth [the] male child, [and] her child [...] caught up [immediately] to God and to his throne without undergoing any pain and suffering and still this would have been more than sufficient for the redemption.

In my other answer I have also stated from Church teaching that since redemption is by blood, a small drop from Jesus as when he was circumcised would have been more than sufficient for the redemption.



Endnote

There is a paragraph in the article that ends:


  We insist on this point because some have misunderstood the whole
  Catholic doctrine of Redemption as though it were based on a cruel and
  vindictive concept of divine justice. (my emphasis)


I thought perhaps this is what may have prompted the OP's question. I would encourage the OP to read the entire article but I would also add that whichever was the things of God are looked at, man is always faced with a mystery.



Further Reading

Redemption - Home > Catholic Encyclopedia > R > Redemption.