Has Yeshua reigned on David's throne for the last 2000 years? No. God's throne, yes, but that's a separate issue--David's throne is over Israel, not in Heaven. Yet, though there is a break, David's line and dynasty are preserved forever in Yeshua the Messiah, who will return and rule over Israel and the world as promised. Has that break annuled the promise? No!
Now, just because the sacrifices have been interrupted (as indeed they were about to be when Jeremiah penned this prophecy), does that mean that there are no Levites who could offer sacrifices? Not at all. They only need a proper Temple and a cleansing ceremony to begin--indeed, there is already a group that has been trained for just such service in Israel.
In other words, a pause or a delay in the fulfillment of God's promise due to Man screwing up does not annul the promise.
But just as Yeshua must someday reign forever on David's throne to fulfill God's promises to David, the Levites must someday minister before the Lord forever as well, or else the Word of God is broken. Ezekiel 40-48 describes just that happening, and even describes how the Prince (the Messiah) will participate.
Notice how far you're having to go to avoid the plain meaning of this text! You don't even offer a reasonable interpretation; only an easily-overcome objection! Who was it who said when you don't have anything on your side, just argue? Premillennial eschatology can accept that God means what He says and says what He means, but yours must constantly seek loopholes or else ignore 3/4ths of Scripture (declaring it "old") in order to preserve your traditional views!
Actually, yes. Jesus has been reigning the throne of David.
"Men and brethren, let me speak freely to you of the patriarch David, that he is both dead and buried, and his tomb is with us to this day. Therefore, being a prophet, and knowing that God had sworn with an oath to him that of the fruit of his body, according to the flesh, He would raise up the Christ to sit on his throne, he, foreseeing this, spoke concerning the resurrection of the Christ, that His soul was not left in Hades, nor did His flesh see corruption. This Jesus God has raised up, of which we are all witnesses." (Acts 2)
The throne of David was representative of the reign of Israel. That is precisely where Jesus sits right now, reigning and subduing the nations, Israel included. As Jews and gentiles come to Christ by the power of the Holy Spirit, they testify of Christ's reign.
That probably won't satisfy your hermenuetical presuppositions, but if does fit the facts.
Those who claim that Jesus will physically sit on "David's throne" can't possibly account for the fact that David's physical throne does not exist, and has not existed for millennia.
In other words, a pause or a delay in the fulfillment ...
Notice how far you're having to go to avoid the plain meaning of this text!
You've just confirmed that the "plain meaning of the text" is not so plain, and requires a certain amount of "spin" (in you case by inventing a "pause or delay") no matter what your persepctive.