Is Google Docs an appropriate tool to write a PhD thesis?
My prefered tool would be markdown / latex, but my primary supervisor really does not like it, to the point that collaboration with this involves printing the document, hand writing notes on it and me transcribing these notes to the original. 


This is an excellent way to collaborate!


  We are currently preparing a paper on the initial work of my PhD, and this is being done successfully through google docs. Would this be an appropriate tool to use for my whole thesis?


Google Docs is as good as Microsoft Word, LibreOffice, etc. Personally I think they are all awful for academic writing.

Perhaps your supervisor can use Overleaf? You can then use LaTeX/git in the usual way.