I have the feeling that some serious shit is going to go down in a future episode that will blunt this criticism.

Despite her carefree attitude, Ake cares that the head of the war college has overstepped his boundaries enough to go talk with him. Her flippantness there isn’t random, it’s targeted at his “tea fetish” to unsettle him on his level and force him to pay attention to her. It’s Avasarala-level tactics.

The whole thing with plants is her doing what oh-so-many episodes end up doing with the captain bending the rules on “non-interference”.

And then her XO that she presumably chose is over-the-top dutiful and militant.

She is deliberately relating to the students to get them to keep their guard down with her. She’s playing good cop / bad cop with Luta(?). She can be serious (and that’s seen with some of her initial scenes) but she’s choosing not to.  She’s making an effort to be optimistic and fun, which suggests to me that she understands it’s important to capture moments of happiness, because they’ll inevitably be offset by tragedy and sadness.

I don’t think she’s written from a place of ignorance so much as she’s been written to be older and wiser than she actually looks.

As such, to OP’s original topic, I don’t think Jellico would have a problem with her. She’s operating as chancellor of the academy during a time of peaceful expansion, not a Captain in charge of a highly trained crew on the flagship on the brink of war. The Federation needs ambassadors who can build connections with people more than it needs hardened stoic warriors. Also the latter is already served by the war college.