I would guess that they trained AlphaGo from many thousands of hours of match footage. Writing a computer vision script to segment / extract the data may cost cycles as you say, but would save many human hours by eliminating the need to re-watch the footage and literally type out state information for each move.

Alphago was actually trained directly on game state (plus some extra computed state like "how many liberties will I have if I play this move" or "will I win this ladder"). A huge number of pro games (and countless amateur games) are available on servers like KGS in a nice computer-digestible format.

Again, that doesn't make sense because the moves have almost certainly already been typed out by somebody. It's the same in chess. There are databases containing millions of games.

I'm not too familiar with Go, but in chess we have millions of text-format games and very little real-time footage.