Thank you for asking this sincerely. Everything depends on who "him" is in this verse, so let me start there. In the classical Jewish reading, the servant of this whole passage (Isaiah 52:13 through 53:12) is Israel. Rashi opens it with "My servant Jacob, the righteous within him" [S3], and explains that this prophet regularly speaks of all Israel as a single man, just as in Isaiah 44:2 [S4]. Ibn Ezra reads "My servant" as every Israelite in exile who serves God, "as in 'Israel My servant'" [S5], and Radak states plainly that the passage was said about the exile of Israel, who is called "My servant" exactly as in Isaiah 41:8 [S6]; see also Isaiah 49:3, where the servant is named outright.
Who, then, is speaking in "our sicknesses he bore"? Rashi says these are the words the nations will say at the end of days, when they see Israel taken out of their grip and out of the suffering it bore, exiled from "the land of the living," the Land of Israel [S7]. They had assumed he was hated by God, "and it was not so"; rather, he was being chastened with sufferings [S8]. Radak adds that this is the nations' own realization at the time of redemption: the faith Israel held was the true one, so the sickness and pain that should have come upon them had fallen on Israel instead [S9].
Isaiah 53:4 Yet it was our sicknesses that he bore, our pains that he carried; and we had thought him plagued, smitten by God and afflicted.So what is the "illness" of our verse? Not a teaching that God sends disease to end a life. Ibn Ezra and Radak both say the "pains" and "illness" of this passage are the sorrow of exile [S10][S11], and Ibn Ezra on our verse puts it in one phrase: "He desired to crush him, to chasten us in exile" [S13]. Rashi reads "God desired to crush him, He made him ill" as God's wish to bring him back to the good, and "asham" as a kofer, the amends a person gives to the one he has wronged [S12]. Radak: in his pains and troubles in exile we can see nothing but the will of God, for he held fast to the Torah of truth [S14].
And notice how the verse ends: "he shall see offspring, he shall prolong his days." That is the future of a people, not a deathbed. Ibn Ezra: when he confesses his guilt, God will multiply his children and lengthen his days so that he and his children see God's salvation [S13]. Radak links it to Isaiah's own promise that "the days of My people" will be "as the days of a tree" (Isaiah 65:22), and notes that at this point the nations' words end and God's own words begin [S14]. So yes, people do die of disease, but that is not what this verse teaches. "He made him ill" here is the affliction of a nation refined in exile and then restored, with children and long life as its reward [S1][S2].