SharePoint doesn't know where to store document when using template from content type

I have created a content type on tenant level that contains a template word document. Have created a document library and enabled "allow management of content type". Have added the template content type. And as expected, I have this template in the "new" button in the document library.

When using Word Online, everything works fine. I create a new word document based on the template, and I can save it with a new file name and it will be stored in the document library.

The problem comes when I configure the document library to use desktop apps. When I create a new document based on the template, the Word App opens. It's all fine, until I try to save it. It doesn't know where to store the file, so it just suggests the root of OneDrive. I'll have to browse to the document library to store it the correctly. Not a good solution.

Is it possible to make Word App store the new file the right place without asking the user where to store it?

If I create a new template by using "Add Template" from the "New"-menu, things works better. If I (still) configure SharePoint to open documents in Desktop Apps, the file will open as Word Online the first time - and that fixes the problem. The next time I open it, it opens in Word App.

However, the problem with this is that if I need to update the template in the future and I use the template in 20 different document libraries, I'll have to remember to update the template in 20 different places.

If there a solution for this?