I'm really hoping the author is thinking so hard about this, but the political game really explains why he was removed from succession.
The first thing you need to understand is that there are always factions within kingdoms vying for their interests. The leading cause being the overthrow or curtailing of the king. Spend any time reading up on the history of just about any country (though I recommend England and Sweden for this matter) and you'll see plenty of it.
With that out of the way we can talk about the background political situation. The protag's father is set up to be, not just a duke, but at the very top of the social order, beneath the king. This means that he is most likely the most powerful duke in the kingdom, widely respected and the leader of the king's supporters. The marriage between his daughter and the crown prince was to cement the bond between the two houses and provide a strong bulwark in the kingdom.
So what happened instead is that the crown prince acted dishonorably, broke off the engagement and banished the daughter. This is bad in three different ways:
1. His behavior shows him to be unready and a weak successor. This is ample fuel for opposing factions and will see claimants that usually would not be considered be given more worth as a 'lesser of two evils'.
2. He destroyed the connection between the duke and the Royal house. The duke simply removing his support would most likely be dire, but he could even join the factions against the king, leading to the steady downfall of the Royal house.
3. The other prince has asked for the daughter's hand in marriage. This shifts the alliance of the duke to outside of the kingdom. This could lead to the dukedom exiting the kingdom to join the other. Hell, we haven't seen the duke having any successors, so it might very well be that the daughter is his successor, in which case the dukedom simply leaves the kingdom in the most boring way possible.
The prince didn't just fuck up. He made the kind of decision that throws a kingdom into a Civil War. That alone should see him removed, but the more pressing issue is that removing his is the only way to deal with the massive injury he's dealt to the country. If he isn't removed, even if the duke keeps supporting the king, there is no way he will support the crown prince upon his coronation. The incident will not be forgotten either and the lacking nature of the successor will plague the future of the kingdom. So the prince gets removed to avoid a civil war.