I feel like the men are being all yes men to her and going along with whatever she says.
They're used to it and used to seeing the princess be so adaptable to war and things that they think it's going to be something she can endure.
I like that she's speaking up but this isn't the time nor place for that.
She spoke of how she is the lady of the territory and as a lady in that time period with no fighting capabilities, she should be tending to the territory and improving the way of life there for their people. Being a woman who takes care of home and the people are her is strength too. That's strong and growth too. Everyone has their roles to fill with everything and in this case hers is at the castle.
This man Riftan has been to war many times and he and his men are used to the perilous conditions (as they mentioned) they'd face on the journey.
I don't know what she's trying to prove by being so insistent on attending the expedition with them.
If she does and gets hurt then what? I think that Riftan in this case has every right to Check her just as she had to check him before on his bullshit too.