The online Bible that I use gives a note as follows to the verse below:-
Matthew 15:24 But he answered and said, I am not sent but unto the lost sheep of the house of Israel
(The Lord Jesus had come into the world to die for the sin of the whole world, but He had also come as Israel's promised Messiah. His seeming harshness to the Canaanite woman is best understood as not only a test of her faith in the God of Israel, but also as a means to show His disciples that Gentiles also were included in God's plan, and that they too could have saving faith.)
Does that explain your last point?
Matthew 15:24 But he answered and said, I am not sent but unto the lost sheep of the house of Israel
(The Lord Jesus had come into the world to die for the sin of the whole world, but He had also come as Israel's promised Messiah. His seeming harshness to the Canaanite woman is best understood as not only a test of her faith in the God of Israel, but also as a means to show His disciples that Gentiles also were included in God's plan, and that they too could have saving faith.)
Does that explain your last point?