What an insightful question! You have found (at least partially) the key to Annie's own method! Oh my, you have to read between the lines to find it, though. Quite honestly, I have always overlooked that part of the play until I read your question. I think the audience gets caught up in the fact that Kate is losing all hope and forgets to listen to the important tidbit that Annie imparts here. Let's look at it:
Dr. Howe did wonders, but--an ordinary child? No, never. But then I thought when I was going over his reports--(She indicates the one in her hand)--he never treated them like ordinary children. More like--eggs everyone was afraid would break.
In simple terms, Dr. Howe's flaw according to Annie was that he never treated his patients like "ordinary children." He took too much care with them, as we might cradle an egg. This is what keeps them from reaching their true potential. They aren't eggs, they're children. They don't need to be coddled and pittied like Helen is being coddled and pittied at home, ... teachers and doctors need to have higher expectations that Helen would be expected to reach. In this way, her true potential would be met. (Not to mention Helen would be punished, sometimes in some creative ways, just like "ordinary children" are.) Thank God for Annie!