In Ramayana, at each and every point it teaches us Dharma, nothing else but Dharma.
Was sending Sri Ramji to the forest Dharma?
Was fulfilling a vow to the wife and not to the citizens of the kingdom Dharma?
Was sending Sita to the jungle Dharma?
Many such questions may arise in the mind while we go through Ramayana.
Let us assume what would have happened if Dasharatha broke the promise given to Kaikeyi. Nobody would have known about the promise and Sri Ramji might have become the King. But Satyam can never be hidden. Later, when the people of Ayodhya will come to know that the king broke a promise to his wife, wouldn’t they think – “What a king is Dasharatha who can’t even fulfill his word to his wife? A person who breaks his promise, and that too to his wife, can he be trusted?” Faith in the king would have been lost and so would the peace in Ayodhya. The king has to do Dharma towards his people first and then comes his Dharma towards his family. That is why Dasharatha couldn’t say anything against Kaikeyi. Same in the case of Ramji when Sitadevi had to be sent to the forest.
Where there is Dharma, there is peace. In Ramayana, Dasharatha did Dharma. Hence there was no war within the family. In Mahabharata, Dhrutarashtra did not do Dharma and protected his sons who were doing Adharma. Hence there was war within the family.