How do you approach the problem of a champion that performs too well in soloqueue but very badly in professional (or at least very coordinated) play? And vice versa?

This is a perennial puzzler for our live design department. You should go talk to FeralPony as this falls neatly within the set of things his team works on.
As a champ designer, in short you think in terms of skill ceiling and skill floor. Skill floor is: "how much are you helping your team if you just mash your buttons senselessly?" vs skill ceiling being "what can Faker do on this champion?" In 2014, we over-indexed on high skill ceiling (Azir, Kalista, Rek'Sai); before that we over-indexed on skill-floor (thinking in terms of "good first play experience" and "baseline contribution"). We're currently trying to find a happy middle ground.