Well, I don't know anything about the creative team behind John Carter. I know that the kind of respect that Peter Jackson had for LotR or Nolan has for Batman is nothing to be taken for granted. Usually, it's just an attempt to leverage an IP, and it gets distorted and retrofitted into something whose primary purpose is to target a demographic. Anything that doesn't serve that purpose gets cut.Ah. You see, there's a middle ground between being slavishly loyal and all things that are included are so for specific modern reasons. It's called "trying to be generally faithful". In this mode, it is the *changes* that are what you have to justify, rather than the things you keep. Did they have a good reason to remove Woola? No? Then they try to do a good job of portraying the spirit of the character.
I'm sure kids will like Woola, but in the film, the creature plays roughly the same role as in the book - telegraphing the fact that John Carter, while maybe not perfect, is a basically good human being. It's a fairly standard trope - the love and loyalty of animals (and usually children) comes to those of good heart. So, while Carter is spending effort claiming that he's mercenary, and only after his cave of gold, Woola's loyalty tells us otherwise.
But your reasoning behind Woola's inclusion seems valid. Movie protagonists tend to be cast into the roles of one of Dorothy's companions from "The Wizard of Oz". That gives the character some blatant personal flaw to overcome by the movie's climax. In this case, I guess we're supposed to have a Tin Man who needs to find that some things (and people) are worth fighting for.