Reducing the emphasis on thinking out-loud every single one of your thoughts might have helped. I'm not the type of person who can verbalize everything and moreover, it's too slow and prone to errors. Perhaps I should have fought back and asked for the interviewer to be quiet and also let me be quiet for a while.

Also, the absolute requirement that any of your workings should be done only on the shared Google doc seems like an overkill. There are a lot of times when I wanted to quickly note down / draw something on the notebook I had but couldn't.