Besides Ace from Classic, it's hard to see much of a story arc with any of the characters.
You'll have to rewatch City of Death and then watch the initial Clara episodes. It feels like a rip off of the same story to me.
Scaroth was consciously linked to his splinters. And his splinters existed across basically one locale. (Earth, possibly just Europe) In practice he could even have been essentially an immortal singular being, able to interact with his own mind backwards in time.
Clara initially could have been no more than Gwen from Torchwood, or Martha’s cousin at Torchwood tower, or Belinda Chandra and Mundy… etc. That she wasn’t was a plot point.
Clara was not a splintered person, she was inserted into the Doctors Timestream (which is another reason why the TC stuff makes no sense) and we found that out after many many stories in an arc, and she essentially became many different people reoccurring across time and space.
It may feel that way to you, and by the rules of the twenty-first century, hey, cool — but it’s observably not the case.
Edit: as to your other point, Turlough goes from coward acting out of fear being compelled to kill the Doctor, to becoming increasingly brave, the Doctors friend and leaves to save his people. Tegan goes from unwilling companion to wanting to travel when she rejoins, but then leaves when she again feels the toll of the deaths around her. Leela is an overt Pygmalion reference, and goes from superstitious ‘savage’ stereotype to more ‘rational’ and also less inclined to stab first and ask later. Romana was a straightforward Time Lady, used to and happy with that ‘upper class’ existence, but refuses to return to Gallifrey because she wants to be out in the universe helping people. Basically, her arc is a mini version of the Doctors own origin. She even dresses like him after her regeneration.
Lots of companions have growth and arcs, even in the classic series, because they represent an opportunity for writers that wasn’t present in the main character. Though even Doctors have arcs, usually compressed in once they know the actor is leaving. Pertwee’s Doctor is probably the best example of character development, and it’s very apparent by Planet of the Spiders. He used to want to get away from Earth and Unit, but by the end, the Tardis brings him ‘home’.