Isaiah 53 is not just about sin. Much more so, it is certainly not just about disease.
It is a picture of Christ on the cross, so all history leads up to this and all the future feeds back to this. The whole 'curse' of Adam was placed upo Him, and from Him the whole 'blessing' of Abraham is made. This is nothing less than the fall and return to paradise. So in a sense all salvation is here in its primitive kernel.
Even Rabbinic sources before Christ took the Messiah to undergo physical suffering and that would include the healing of physical diseases, not just sin. This can be seen from the Talmud identifyng th Messiah as the 'leperous one'.
Babylonian Talmud (Sanhedrin 98b): "Messiah ...what is his name? The Rabbis say,' The leprous one'... for it says, 'Surely he has borne our sicknesses' etc. (Isa.53,4)."
From the New Testament view:
16When evening came, many who were demon-possessed were brought to him, and he drove out the spirits with a word and healed all the sick. 17This was to fulfill what was spoken through the prophet Isaiah: “He took up our infirmities and carried our diseases.” (Mathew 8:16-17)
This makes it sufficiently clear that these words include demon possession and illnesses of body.
24He himself bore our sins in his body on the tree, so that we might die to sins and live for righteousness; by his wounds you have been healed. 25For you were like sheep going astray, but now you have returned to the Shepherd and Overseer of your souls. (1 Peter 2:24)
This also makes it clear that this means he would take away our sins.
So the idea seems to be that that Jesus is the great Messiah, the Healer, Comforter, Sin Bearer and killer of all; sin, hell, sorrow, death, and sickness, through his atonement. This does not mean all sickness and death is removed now, for salvation in history had developments and the full achievement of Christ’s work is in the future. All sorrow and suffering and sickness will not be removed until his future return to judge the living and the dead. However, when God met mankind, in the God-Man, it was impossible for the bizarre situation not to drive all sickness and demon possession away. The same will be true when he returns in glory:
16Never again will they hunger; never again will they thirst. The sun will not beat upon them, nor any scorching heat. 17 For the Lamb at the center of the throne will be their shepherd; he will lead them to springs of living water. And God will wipe away every tear from their eyes.” (Rev 7:16-17)
This is the final dimension of Isaiah 53.