“The Jews don’t need anyone from the House of David - they need Levites so they make the appropriate sacrifices on the Mount.”
The long-promised Messiah (Moschiach) who will bring peace to the world, etc., etc. (he’s got a BIG job ahead of him) will be a man, and specifically a man descended through his father, grandfather, etc., etc. from King David, through David’s son, King Solomon. So, ultimately, YES the House of David is necessary.
The Messiah has already come. We just need to be ready for his return. No one knows the day, but the sooner, the better.