fulfilled under Tiglath-pileser.
Almost true, but only half-way there. Never forget this:
Ec 1:9 The thing that hath been, it is that which shall be; and that which is done is that which shall be done: and there is no new thing under the sun.
There is an earthly course, and there is an heavenly course. Carnal and spiritual; natural first, then spiritual. All of these earthly things you're being shown in the old covenant are just prophetic carnal pictures of spiritual things to come. For instance, the Exodus: this earthly event is but pictures- foreshadow- of the TRUE exodus to come by Jesus Christ. First the natural exodus by the hand of Moses, then the spiritual exodus by the hand of Jesus. These are all earthly pictures- prophetic of the spiritual work to come.
In the same way Tiglath-pileser is also just a picture of someone else to come. Someone who is going to be sent by God to execute the indignation of the Lord against those who transgress the covenant (as was the Assyrian; Isa 10). Moses was a picture of Christ; the Assyrian is a picture of the son of perdition spoken of by Paul, and the false prophet spoken of by John- commonly known by most as "the antichrist"
In the prophesies, Nebuchadnezzar King of Confusion (Babel/Babylon) is the "chief" picture of the Destroyer; but the Assyrian is also a picture of the same man of sin to come, but the focus here is on the dominion of Israel whereas the focus of the King of Confusion is the sanctuary of Judah.
And as it was by the earthly Assyrian that earthly Israel was judged, so it is by the spiritual Assyrian to come that the spiritual Israel will be judged. And not only, but the world entire. Isaiah isn't speaking of the desolation of Damascus carnal war as many now wrongly understand- but he is speaking of the spiritual desolation of spiritual Damascus to come at the hand of the spiritual Assyrian- who is on his way as we speak.