As king candidates these people are currently failures.
They haven't even put any effort into thinking why the previous king set things up in this manner.
Like two of them when the FL says that they should get along because they are from the same royal family are dismissive of her, the third one sees his duty as a knight more important than those meetings.
Each day this goes on the populace becomes more uneasy, and the longer it goes on, the more precarious the position the next ruler will start on.
Each of these people are valuable to the kingdom, and have their own important contributions.
The important thing here, however, is that they are at odds with one another.
If they thought about the intentions of the late king, they would realize that a situation where the three other people must vote for the last means one thing: He wanted them to put aside their differences. He wanted them to come together and unite, and do what is best for the kingdom. He didn't want a situation where one of them is declared king, but the rest don't get along with him/her. These are some of the most influential people in the kingdom, and the late king does not want them to be at odds. He doesn't want their bickering to damage the management of the kingdom.
So he forces a situation where they must learn to work together, and get over their differences, fast.
However, they are currently failing hard.
Two of them being at odds for every little thing is unacceptable. One deciding his work in the Knight Order is more important than trying to resolve who the next king of your country and leader of your people is unacceptable.
Francesca also is lacking. Her parents sheltered her way too much. They allowed her to become a recluse at a young age, and she is the future of their house. After she took a heavy blow to her self-esteem to the point of wanting to avoid all social contact as a child, they allowed it, they enabled that all the way to adulthood . All things like that does, all letting her hide away for so many years does is reinforce and make worse her self esteem issues. All it does is make her unable to interact with new people. Cause her to not just stagnate, but regress socially, and isolate her from any potential friend or peer group. This type of "kindness" is wrong and destructive.
As such even if, by chance she knows a lot about foreign countries and languages from her reading, even if she has honed a sharp mind, she is years behind others on social development, and hasn't networked with any of the other nobles. She properly express herself and show off her good qualities to the others, who look down on the fact that she is even a candidate at all.
She has no powerbase besides a king that has already pasted and a family that has let her hide away, while totally lacking the necessary skills to build said powerbase.
It is a good thing that this ploy of the late king also forces her to socialize, and puts her in contact with three of the most powerful people in the kingdom for networking.
However, if the king intended for her to be a king candidate, he really should have done things that would have forced her to slowly move out of her shell far before she hit 18. Though who knows maybe he tried some things before and was rejected. Or worse, her family hid it from her, just like they hid knowledge that she got sent marriage proposals.