Everything here is true.
Truly the Israelite remained/remains. Levites are Israelites, Judaites and bejaminites are Israelites too. But Israel (as in the "the complete kingdom of Israel") ceased to exist.
Recall that Herod (an Edomite) was on the throne of Judea, not Joseph (a true descendant of David). This was the backdrop prior to Messiah's appearance. Next, recall what they asked Messiah after His resurrection just before He ascended to heaven...
Acts 1:6-8 [brackets mine]
When they therefore were come together, they asked of him, saying, Lord, wilt thou at this time restore again the kingdom to Israel? And he said unto them, It is not for you to know the times or the seasons, which the Father hath put in his own power. But ye shall receive power, after that the Holy Ghost is come upon you: and ye shall be witnesses unto me both in Jerusalem [capital of Southern House], and in all Judaea, and in Samaria [the former capital of Northern House, by Omri], and unto the uttermost part of the earth.
So even after the Messiah's resurrection Israel wasn't restored...you make a great point though because recall that even Solomon referenced the promise to David in 1 Kings 8:25 (before Jeremiah)
25 Therefore now, Lord God of Israel, keep with thy servant David my father that thou promisedst him, saying, There shall not fail thee a man in my sight to sit on the throne of Israel; so that thy children take heed to their way, that they walk before me as thou hast walked before me.
...but I think what I've underlined in the passage is the key. The throne of United Israel must first exist. As long as that throne exists David will have descendant on it.