Hi martin.coupal,
First, my pleasure to assist you.
As per your description, please don’t worry, we can work together to narrow down and resolve the situation.
Based on my test result in IOS device when open blank Word document file and click on folder button>select SharePoint site, there I can see all SharePoint site listed.
Since you already tried, sign-out/in account and uninstall, reinstall but issue still persist, First, please make sure your Office 365 app is up-to date in your IOS device,.
Also, please try to use another IOS device and check the outcomes, it will help for us to narrow down your issue.
I appreciate your understanding and stay safe!!
Best regards
Waqas Muhammad