You just described me haha. In my Google phone screen, I fumbled around on solving an easy-moderate problem - so much so that when it ended, I felt that there was no way even I would hire me. And once it ended, I wrote the code for it in probably 10 minutes. Also, arranging it so that it was the first in the loop did not help either. The only good thing about the process was that the recruiter was nice about the whole debacle and did not make me feel too embarrassed.
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.
