Having worked on some code very similar to this, showing the computer's best moves would be quite artificial. Here's some thoughts as to why that is:
1. The computer can discard all its current best ideas and flip through new ones so fast, it would be a flickering blur to humans.
2. Even if we put a speed limit on it, the move being considered is itself the result of considering a lot of slight variations.
3. The ability to _articulate_ in a human language what makes the move nice is itself a "hard problem" closely related to natural language processing.
4. Even just having some color codes or symbols and grouping related ideas has some serious problems: now the visualization is pretty technical to begin with, the computer is still able to memorize and compare moves at an unbelievable rate, and it's still fundamentally not the same as the method Go masters use to find a solution.
I can back this up a bit. I created a strong Chess engine variant and had it visually show what the computer was considering as moves with strengths as color strength. It would even show what it considered your best counter-moves.
Even with all that thinking output on the screen, the computer would still soundly beat myself and another (intermediate) player.
Here are some screenshots to illustrate what I'm talking about:
Regarding the first one, I don't think this would be a serious issue. It shouldn't show what move its considering, just the current best option. It would probably converge on a good move within the first second. Running the free, quick analysis from chess.com on my Iphone has a great visualization that rapidly updates the computer's scoring of the current position and shows what it thinks is the best move, as well as pointing out any previous moves it thinks are mistakes/blunders.
I'd love to see one day a live commentary, with an extra window showing what computer is thinking at the moment.