Hi Brebre,
Well that right there is the issue. That link is for upgrading within the same version number. That guide is for when the build numbers are the only thing different. In our case, the version numbers are different too. As written in the top of the page: “If you are updating to a new point version (2.0.0 to 2.0.1), then you need to follow the instructions for Updating to a New Version of ExpressionEngine.”
Following those steps should fix your white screen issue 😊
Let us know how it works.
Having a single click update comes with a lot of risks. It is something we have looked into but not ruled out. You are welcome to make a feature request for it!