Actually, it is, once obliquely, and once directly. It's almost certain that the Ark was in Israel as late as the reign of Hezekiah. In 2 Ki. 19:15 and Isa. 37:16, we see the king praying, "O LORD God of Israel, who dwells between the cherubim . . ." Such a title is always used of God in reference to the Ark of the Covenant, per Exo. 25:22, 1 Sa. 4:4, 2 Sa. 6:2, 1 Ki. 8:6-7, and 1 Ch. 13:6.
Further, in 2 Ch. 35:3, we see Josiah saying to the Levites, "Put the holy Ark in the house which Solomon the son of David, king of Israel, built. It shall not be a burden on your shoulders. And serve the LORD your God and His people." Therefore, Shishak could not have captured or destroyed it, since its location was known to the Levites centuries later, though they had evidentially removed it from the Temple at some point before Josiah's reign. Since Josiah knew that they had it, it seems unlikely that it had been missing for centuries, but had been removed relatively recently.
Therefore, the most likely explanation for the Ark's absence is that it was removed during the reign of Manesseh, Josiah's predecessor, and it was probably at this time that it was taken to Elephantine Island, where as you said it remained for several centuries in what was apparently a scale copy of the Temple in Jerusalem (we even have papyri asking the priests in Jerusalem how to properly conduct sacrifices--which only makes sense if those on Elephantine had the Ark with them). When Josiah took the throne and led Israel in repentence, he asked the Levites to bring it back. The Bible doesn't say that they did so.
Shortly after that, we see Pharaoh Neco coming up to go to war with the Assyrians at God's order (2 Ch. 35:20)! But Josiah continued to go after Neco in disguise until he was slain in combat. Why would he do this against someone who was siding with him against Assyria at the command of the God of Abraham, Isaac, and Jacob? While the Bible does not come out and say so, to me it seems likely that he was attempting to capture Neco to ransom him for the return of the Ark.
As for why the Bible doesn't just come out and say that the Ethiopians have the Ark, I suspect that was to safeguard the Ark.
I agree that the Ark wasn't in the temple in the time of Josiah. I'm sure the Levites were as puzzled as the king was as to the Ark's location -- and if not, their predecessors may have hidden it. But these later references aren't references to the Ark's having a known location, the 2 Kings and Isaiah citations refer to the Lord, rather than the Ark.