And we have a "sands of the sea" problem, if the northern kingdom was just simply assimilated into the surrounding people at the time.
There is one possible fulfillment of the prophecy.
Matthew 10;5 These twelve Jesus sent forth, and commanded them, saying, Go not into the way of the Gentiles, and into [any] city of the Samaritans enter ye not:
10:6 But go rather to the lost sheep of the house of Israel
That seems pretty clear. I can't see it referring to the remnants of Judah there at the time. They had already been preached to, and the disciples were themselves Judah, or Benjamin, or Levite. Where, then, did the disciples go?
Any other reasoning that the prophecy was fulfilled demands that the House of Israel returned to the homeland in vast (sands of the sea) numbers, which there is no evidence of.
If the Jews of today are sole surviving remnants of all of Israel and Judah, there are only 4 or 5 million, and not a foretold blessing to all nations, except for some scientific inventions here and there.
Prophecy fulfilled doesn't ring true, or fit all of the necessary specs in the Hosea prophecy or the sticks prophecy. To interpret them in the way you do, one must labor numerous assumptions and discount population numbers.
In the absence of direct, agreed upon and compelling evidence to the contrary, the plain wording of the prophecy has to be assumed.