Keep in mind that the Tyrells were already plotting to manipulate Tommen by that point. It was mainly to rid themselcss of Joffrey, but they certainly weren't going to let things go to chance on hiw he'd turn out.
It would have been interesting, but, Tommen was a blank slate at that point, and with multiple parties vying to influence him, its hard to tell how it would turn out. Especially since he was so young that he'd definitely go through phases over time.
It's interesting to remember that a the start of the books, when Ned first decided to go down to KL, was that part of the plan was to bring Bran, in the hopes he'd build a friendship with Joffrey.
Even with good intentions, there's a big desire to get into the influence of kings and heirs. Tywin would have a huge impact on him, but, his relationshio with his own kids suggest that he's fairly insensitive to the desires of the individual, and a moody teenager would likely tune out the old man eventually.