not did she appear to be in any danger of being thrown off the ship once she was on and I think that could have been a running thread.
Thrown off where? The ship was far beyond the outermost starbase or Federation outpost. One way or the other, she was committed to being there for the whole mission.
And Trys made the decision to start being a better officer as soon as she was sent back from the Rhea
attack. From the moment she met Picard, she was doing her best to fit in; the only officer she had any serious clashes with was Kadohata, and as I said, that was specifically due to her issues with her mother rather than any sort of generalized insubordination or ineptitude. So it wouldn't have made sense for there to be a "running thread" about her almost getting thrown off the ship (figuratively or otherwise). I say she was a poor officer at the start, but that's because she wasn't really trying her best. She was trying her best on the Enterprise