The number of descendants of King David through the patrilineal line by Jewish mothers might be much smaller than one might imagine. Persecutions (some targeting Davidic descendants) and the Ashkenazi ‘bottleneck’ some 30 or so generations ago increase the likelihood of fewer Davidic descendants.
But even so, the number might still be in the tens of thousands out of 12 million or so Jews.
True if you’re only talking about those eligible to be the Messiah. However, the number who could - in theory - be descended from David is likely a pretty large percentage of the combined European and Mideastern population (including, of course, those of such descent who migrated elsewhere).
There will be some percentage of Africans, via Solomon’s son, Menelik (from the Queen of Sheba), plus still more from those Jews escaping to Africa after the destruction of either the First or Second Temples.
Yes, the Ashkenazi bottleneck is a big problem - if you’re predisposed to think that the Messiah should be a European Jew or descended of them. At least statistically - none of us could possibly say if, how, or to what extent G-d decided to protect David’s European descendants during that time period.