The way I see it the puzzles serve two vital purposes; 1) They make you examine the environment and immerse yourself further in the world and 2) they build a relationship between the player and Yorda.
I think the puzzles brought the castle to life. The windmill could have been just a non-interactive prop. You'd look at it and then you'd go on your way to the next cutscene. Never stopping long enough to enjoy the world, listen to the wind howling or the sea crashing against the rocks 200 feet below.
In past games I've played with co-operative AI characters I usually curse at their lack of proper path finding and slow reactions. The scientists in Half Life have been subject to many ritual executions by my hand. But in Ico the interaction between the characters is linked in such a wonderful, dynamic manner that I really began to care about Yorda. The act of helping her over obstacles, reaching out to catch her and saving her from monsters made me feel protective. I could never willingly do her wrong.