Becoz the 1st prince is not a child from king & Queen. I can said he is illegimate child or he is a child from father mistress who come from Hariz which im not sure but i think she is commoner. Technically his brother not pure royal or legal. Usually the child who are not come from Queen will not take the throne except if the king have many wives & have many child &they fight for the throne. About the topic u said 'child of mixed blood' is not the main topic. To easier for u to understand, We can put their marriage as a political marriage between Hariz &Gombuz(idont remember to spell) to reconcile them from continue be enemy & war. If they have a child, it will be a strong bond between this two country becoz with this heir they cannot harm each other country. The child so valuable becoz the father is the king from Gombuz while his mother is a princess from Hariz. I hope u will not confused about this.
So a child of mixed blood was supposed to heal everything so WHY couldn’t the first prince have just inherited the throne?