Judgment and remnant
Isaiah has been prophesying about judgment coming upon Israel as a result of her wickedness. Dr. Constable has this to say:
The prophet had just described Assyria cut down like a forest of trees
(10:15-19, 33-34). Likewise, Israel would have only a remnant left after
God finished judging her (10:20-23; cf. 6:11-13). Now he pictured a shoot
(Heb. nezer) sprouting from one of the stumps left after Israel's harvesting
(cf. 4:2; 6:13; 53:1-3; Job 14:7). A shoot would sprout from Jesse's family
Isaiah has been prophesying that Israel would be judged, but that a remnant would survive it. By using the imagery of a chopped tree to describe Jesse, the author relates the judgment coming upon Israel to that coming upon Assyria.
The first wave of the judgment would come at the hands of Assyria. Isaiah's audience would experience that in very short time, hence his focus on Assyria as God's instrument, and the judgment coming upon Assyria as a result of their evil.
Isaiah does not simply leave his audience with a promise of judgment, and then judgment of the instrument of judgment. He (like a good prophet) goes on to prophesy about Israel's Messianic hope.
Then a shoot will spring from the stem of Jesse,
And a branch from his roots will bear fruit.
The language that follows makes it clear that this "shoot" / "branch" is the Messiah.
As God's mouthpiece, Isaiah wants his audience to know that judgment is coming upon Israel, and they (the audience) would experience it at the hands of the Assyrians. With that said, they should rest assured that the Assyrians would get what was coming to them, and the Messiah would one day come through Jesse's line. He would come after the judgment, which should remind them of the hope they had from God of a remnant surviving the judgment.
The focus is on the judgment coming upon Israel, and specifically, that which Isaiah's audience would experience. The reason Jesse's stump is mentioned here (when the Babylonian oppression wouldn't take place until much later) is that it is linked to the post-judgment Messianic hope.